# Ideals as generalized prime ideal factorization of submodules

K. R. Thulasi∗, T. Duraivel, and S. Mangayarcarassy

###### Abstract.

For a submodule $N$ of an $R$-module $M$, a unique product of prime ideals in
$R$ is assigned, which is called the generalized prime ideal factorization of
$N$ in $M$, and denoted as ${\mathcal{P}}_{M}(N)$. But for a product of prime
ideals ${{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}}$ in $R$ and an
$R$-module $M$, there may not exist a submodule $N$ in $M$ with
${\mathcal{P}}_{M}(N)={{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}}$. In
this article, for an arbitrary product of prime ideals
${{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}}$ and a module $M$, we find
conditions for the existence of submodules in $M$ having
${{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}}$ as their generalized prime
ideal factorization.

The first author was supported by INSPIRE Fellowship (IF170488) of the
Department of Science and Technology (DST), Government of India.
MSC(2010): Primary: 13A05; Secondary: 13A15, 13E05, 13E15
Keywords: prime submodules, prime filtration, Noetherian ring, prime ideal
factorization, regular prime extension filtration.
$*$Corresponding author

##  1\. Introduction

Throughout this article, $R$ denotes a commutative Noetherian ring with
identity and $M$ will be a finitely generated unitary $R$-module. The
reference for standard terminology and notations will be [3] and [4].

A proper submodule $N$ of an $R$-module $M$ is called a prime submodule of $M$
if for any $a\in R$ and $x\in M$, $ax\in N$ implies $a\in(N:M)$ or $x\in N$.
If $N$ is a prime submodule of $M$, then $(N:M)=\mathfrak{p}$, a prime ideal
in $R$, and we say $N$ is a $\mathfrak{p}$-prime submodule of $M$. Let $N$ and
$K$ be submodules of $M$. Then $K$ is called a $\mathfrak{p}$-prime extension
of $N$ in $M$ if $N$ is a $\mathfrak{p}$-prime submodule of $K$, and it is
denoted as $N\overset{\mathfrak{p}}{\subset}K$. In this case,
$\mathrm{Ass}(K/N)=\\{\mathfrak{p}\\}$.

Let $N$ be a proper submodule of an $R$-module $M$. Then we have
$\mathfrak{p}\in\mathrm{Ass}(M/N)$ if and only if there exists a
$\mathfrak{p}$-prime extension of $N$ in $M$ [1, Lemma 3]. A
$\mathfrak{p}$-prime extension $K$ of $N$ is said to be maximal if $K$ is
maximal among the submodules of $M$ which are $\mathfrak{p}$-prime extensions
of $N$ in $M$. Since $M$ is Noetherian, maximal $\mathfrak{p}$-prime
extensions exist. A filtration of submodules
$\mathcal{F}:N=M_{0}\overset{{\mathfrak{p}}_{1}}{\subset}M_{1}\subset\cdots\subset
M_{n-1}\overset{{\mathfrak{p}}_{n}}{\subset}M_{n}=M$ is called a maximal prime
extension (MPE) filtration of $M$ over $N$, if
$M_{i-1}\overset{{\mathfrak{p}}_{i}}{\subset}M_{i}$ is a maximal
${\mathfrak{p}}_{i}$-prime extension in $M$ for $1\leq i\leq n$. It is proved
that
$\mathrm{Ass}(M/M_{i-1})=\\{{\mathfrak{p}}_{i},\ldots,{\mathfrak{p}}_{n}\\}$
for each $1\leq i\leq n$ [1, Proposition 14]. Hence, the set of prime ideals
which occur in any MPE filtration of $M$ over $N$ is exactly equal to
$\mathrm{Ass}(M/N)$.

A maximal $\mathfrak{p}$-prime extension $K$ of $N$ is said to be regular if
$\mathfrak{p}$ is a maximal element in $\mathrm{Ass}(M/N)$, and the filtration
$\mathcal{F}:N=M_{0}\overset{{\mathfrak{p}}_{1}}{\subset}M_{1}\subset\cdots\subset
M_{n-1}\overset{{\mathfrak{p}}_{n}}{\subset}M_{n}=M$ is called a regular prime
extension (RPE) filtration of $M$ over $N$ if
$M_{i-1}\overset{{\mathfrak{p}}_{i}}{\subset}M_{i}$ is a regular
${\mathfrak{p}}_{i}$-prime extension in $M$ for $1\leq i\leq n$. In this case,
for each $i<j$,
$M_{i}\overset{{\mathfrak{p}}_{i+1}}{\subset}M_{i+1}\cdots\subset
M_{j-1}\overset{{\mathfrak{p}}_{j}}{\subset}M_{j}$ is also an RPE filtration
of $M_{j}$ over $M_{i}$. Since RPE filtrations are MPE filtrations,
$\mathrm{Ass}(M_{j}/M_{i})=\\{{\mathfrak{p}}_{i+1},\ldots,{\mathfrak{p}}_{j}\\}$
for $1\leq i<j\leq n$. In particular,
$\mathrm{Ass}(M/N)=\\{{\mathfrak{p}}_{1},\ldots,{\mathfrak{p}}_{n}\\}$.

The following lemma gives the condition for interchanging the occurrences of
prime ideals in an RPE filtration.

######  Lemma 1.1.

[1, Lemma 20] Let $N$ be a proper submodule of $M$ and
$N=M_{0}\subset\cdots\subset
M_{i-1}\overset{{\mathfrak{p}}_{i}}{\subset}M_{i}\overset{{\mathfrak{p}}_{i+1}}{\subset}M_{i+1}\subset\cdots\subset
M_{n}=M$ be an RPE filtration of $M$ over $N$. If
${\mathfrak{p}}_{i+1}\not\subseteq{\mathfrak{p}}_{i}$ for some $i$, then there
exists a submodule $K_{i}$ of $M$ such that $N=M_{0}\subset\cdots\subset
M_{i-1}\overset{{\mathfrak{p}}_{i+1}}{\subset}K_{i}\overset{{\mathfrak{p}}_{i}}{\subset}M_{i+1}\subset\cdots\subset
M_{n}$ $=$ $M$ is an RPE filtration of $M$ over $N$.

RPE filtrations satisfy the following uniqueness property.

######  Lemma 1.2.

[1, Theorem 22] For a proper submodule $N$ of $M$, the number of times a prime
ideal $\mathfrak{p}$ occurs in any RPE filtration of $M$ over $N$ is unique,
and hence, any two RPE filtrations of $M$ over $N$ have the same length.

The submodules which occur in an RPE filtration are characterized as follows.

######  Lemma 1.3.

[2, Lemma 3.1]. Let $N$ be a proper submodule of an $R$-module $M$. If
$N=M_{0}\overset{{\mathfrak{p}}_{1}}{\subset}M_{1}\subset\cdots\subset
M_{n-1}\overset{{\mathfrak{p}}_{n}}{\subset}M_{n}=M$ is an RPE filtration of
$M$ over $N$, then $M_{i}=\\{x\in
M\mid{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{i}}x\subseteq N\\}$ for $1\leq
i\leq n$.

Hence, the product of prime ideals that occur in any two RPE filtrations of
$M$ over $N$ is the same. This product is called the generalized prime ideal
factorization of $N$ in $M$ and denoted as ${\mathcal{P}}_{M}(N)$ in [5], and
sufficient conditions for
${\mathcal{P}}_{M}({{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$
were found, where ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ are prime
ideals in $R$ [5, Theorem 2.14].

There may be products of prime ideals that are not the generalized prime ideal
factorization of any submodule of a given module.

######  Example 1.4.

Let $R=\frac{k[x,y,z]}{(xy-z^{2},x^{2}-yz)}$ and
$\overline{x},\overline{y},\overline{z}$ denote the images of $x,y,z$
respectively in $R$. Let $\mathfrak{p}$ be the prime ideal
$(\overline{x},\overline{z})$. Then
$({\mathfrak{p}}^{2}:{\mathfrak{p}})=(\overline{x},\overline{y},\overline{z})$.
Suppose there exists an ideal $\mathfrak{a}$ in $R$ with
${\mathcal{P}}_{R}(\mathfrak{a})={\mathfrak{p}}^{2}$. Then there exists an RPE
filtration
$\mathfrak{a}\overset{\mathfrak{p}}{\subset}{\mathfrak{a}}_{1}\overset{\mathfrak{p}}{\subset}R$
and therefore, $\mathrm{Ass}(R/\mathfrak{a})=\\{\mathfrak{p}\\}$. By Lemma
1.3, ${\mathfrak{a}}_{1}=(\mathfrak{a}:\mathfrak{p})$, and since
${\mathfrak{p}}^{2}\subseteq\mathfrak{a}$, we have
$({\mathfrak{p}}^{2}:{\mathfrak{p}})\subseteq(\mathfrak{a}:\mathfrak{p})$.
Since
$(\overline{x},\overline{y},\overline{z})=({\mathfrak{p}}^{2}:{\mathfrak{p}})\subseteq(\mathfrak{a}:\mathfrak{p})={\mathfrak{a}}_{1}\subsetneq
R$ and $(\overline{x},\overline{y},\overline{z})$ is a maximal ideal,
$(\overline{x},\overline{y},\overline{z})=(\mathfrak{a}:\mathfrak{p})$. Then
$(\overline{x},\overline{y},\overline{z})=(\mathfrak{a}:p)$ for every
$p\in\mathfrak{p}\setminus\mathfrak{a}$. This would imply that
$(\overline{x},\overline{y},\overline{z})\in\mathrm{Ass}(R/\mathfrak{a})$, a
contradiction. Therefore, an ideal $\mathfrak{a}$ in $R$ cannot have
${\mathcal{P}}_{R}(\mathfrak{a})={\mathfrak{p}}^{2}$.

In this article, for a product of prime ideals
${{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$ (${\mathfrak{p}}_{i}$’s not
necessarily distinct), we find conditions for the existence of submodules $N$
of $M$ with
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$. We also
give a necessary and sufficient condition for
${\mathcal{P}}_{M}({{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$.

##  2\. Ideals as Generalized Prime Ideal Factorization of Submodules

######  Lemma 2.1.

Let $N$ be a submodule of $M$ and
${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{r}$ be some minimal prime ideals in
$\mathrm{Ass}(M/N)$. Then there exists a submodule $K$ of $M$ containing $N$
such that
${\mathcal{P}}_{M}(K)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{r}}$.

###### Proof.

Let $N=M_{0}\overset{{\mathfrak{q}}_{1}}{\subset}M_{1}\subset\cdots\subset
M_{n-1}\overset{{\mathfrak{q}}_{n}}{\subset}M_{n}=M$ be an RPE filtration of
$M$ over $N$. Since
$\\{{\mathfrak{q}}_{1},\dots,{\mathfrak{q}}_{n}\\}=\mathrm{Ass}(M/N)$, for
each $1\leq i\leq r$, ${\mathfrak{p}}_{i}={\mathfrak{q}}_{j}$ for some $j$.
Since ${{\mathfrak{p}}_{1}},\dots,{{\mathfrak{p}}_{r}}$ are minimal, we can
reorder ${\mathfrak{q}}_{1},\dots,{\mathfrak{q}}_{n}$ such that
${\mathfrak{q}}_{j}\not\subset{\mathfrak{q}}_{k}$ for $1\leq j<k\leq n$ and
${\mathfrak{q}}_{n-r+i}={\mathfrak{p}}_{i}$ for $1\leq i\leq r$. So using
Lemma 1.1 sufficient times we can have an RPE filtration

$N=K_{0}\subset K_{1}\subset\cdots\subset
K_{n-r}\overset{{\mathfrak{p}}_{1}}{\subset}K_{n-r+1}\overset{{\mathfrak{p}}_{2}}{\subset}\cdots\subset
K_{n-1}\overset{{\mathfrak{p}}_{r}}{\subset}K_{n}=M$

of $M$ over $N$. Then

$K_{n-r}\overset{{\mathfrak{p}}_{1}}{\subset}K_{n-r+1}\overset{{\mathfrak{p}}_{2}}{\subset}\cdots\subset
K_{n-1}\overset{{\mathfrak{p}}_{r}}{\subset}K_{n}=M$

is an RPE filtration. So if $K=K_{n-r}$, then $K$ is a submodule of $M$
containing $N$ with
${\mathcal{P}}_{M}(K)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{r}}$. ∎

Now we show that for a prime ideal $\mathfrak{p}$ in $R$,
$\,\mathfrak{p}\in\mathrm{Supp}(M)$ is a necessary and sufficient condition
for the existence of a submodule $N$ in $M$ with
${\mathcal{P}}_{M}(N)=\mathfrak{p}$. More generally, we have the following
result.

######  Proposition 2.2.

Let $M$ be an $R$-module and ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ be
prime ideals in $R$ such that
${\mathfrak{p}}_{i}\not\subseteq{\mathfrak{p}}_{j}$ for every
$i,j\in\\{1,\dots,n\\}$ with $i\neq j$. Then the following are equivalent:

  1. (i)

$\\{{\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}\\}\subseteq\mathrm{Supp}(M)$;

  2. (ii)

${\mathfrak{p}}_{i}\in\mathrm{Supp}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$
for every $1\leq i\leq n$;

  3. (iii)

${\mathfrak{p}}_{i}$ is minimal in
$\mathrm{Ass}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$ for every
$1\leq i\leq n$;

  4. (iv)

There exists a submodule $N$ in $M$ with
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$.

###### Proof.

(i) $\Rightarrow$ (ii): Suppose
${\mathfrak{p}}_{i}\notin\mathrm{Supp}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$
for some $i$. Then
${({M}/{{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M}})}_{{\mathfrak{p}}_{i}}=0$.
So we get
$M_{{\mathfrak{p}}_{i}}=({{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}})M_{{\mathfrak{p}}_{i}}$.
Since
$({{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}})_{{\mathfrak{p}}_{i}}\subseteq{\mathfrak{p}}_{i}R_{{\mathfrak{p}}_{i}}$,
by Nakayama’s lemma $M_{{\mathfrak{p}}_{i}}=0$. Therefore
${\mathfrak{p}}_{i}\notin\mathrm{Supp}(M)$.

(ii) $\Rightarrow$ (iii): If
${\mathfrak{q}}\in\mathrm{Supp}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$,
then ${{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}\subseteq\mathfrak{q}$,
and therefore $\mathfrak{q}$ contains some ${\mathfrak{p}}_{i}$. So the set of
minimal elements of
$\mathrm{Supp}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$ is
contained in $\\{{\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}\\}$. Since
${\mathfrak{p}}_{i}\not\subseteq{\mathfrak{p}}_{j}$ for all $i\neq j$,
${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ are minimal elements in
$\mathrm{Supp}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$.
Therefore ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ are minimal in
$\mathrm{Ass}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$.

(iii) $\Rightarrow$ (iv): Since ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$
are minimal in
$\mathrm{Ass}(M/{{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M})$, by Lemma
2.1, there exists a submodule $N$ of $M$ with
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$.

(iv) $\Rightarrow$ (i): Since ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$
are the prime ideals which occur in an RPE filtration of $M$ over $N$,
$\\{{\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}\\}=\mathrm{Ass}(M/N)\subseteq\mathrm{Supp}(M)$.
∎

######  Corollary 2.3.

Let $\mathfrak{p}$ be a prime ideal in $R$. Then
$\mathfrak{p}\in\mathrm{Supp}(M)$ if and only if there exists a submodule $N$
in $M$ with ${\mathcal{P}}_{M}(N)=\mathfrak{p}$.

In Proposition 2.2, the prime ideals are distinct. Now we find conditions for
the product of prime ideals that need not be distinct to be a generalized
prime ideal factorization of some submodule.

######  Proposition 2.4.

Let $\mathfrak{p}$ be a prime ideal in $R$ and $r$ be a positive integer. If
$\mathfrak{p}\in\mathrm{Ass}({{\mathfrak{p}}^{r-1}M}/{{\mathfrak{p}}^{r}M})$,
then there exists a submodule $N$ in $M$ such that
${\mathcal{P}}_{M}(N)={\mathfrak{p}}^{r}$.

###### Proof.

Let $N=\\{x\in M\mid({\mathfrak{p}}^{r}M:x)\not\subseteq\mathfrak{p}\\}$. Let
$x_{1},x_{2}\in N$ and $u\in R$. Then there exists
$a_{1}\in({\mathfrak{p}}^{r}M:x_{1})\setminus\mathfrak{p}$ and
$a_{2}\in({\mathfrak{p}}^{r}M:x_{2})\setminus\mathfrak{p}$. Then
$a_{1}a_{2}\in({\mathfrak{p}}^{r}M:ux_{1}+x_{2})\setminus\mathfrak{p}$, which
implies that $ux_{1}+x_{2}\in N$. Hence $N$ is a submodule of $M$. Since
$\mathfrak{p}\in\mathrm{Ass}({{\mathfrak{p}}^{r-1}M}/{{\mathfrak{p}}^{r}M})$,
there exists $x\in{\mathfrak{p}}^{r-1}M$ such that
$\mathfrak{p}=({\mathfrak{p}}^{r}M:x)$. This implies $x\notin N$. Therefore
$N$ is a proper submodule of $M$. Also, $N\supseteq{\mathfrak{p}}^{r}M$.

We claim that $\mathrm{Ass}(M/N)=\\{\mathfrak{p}\\}$. Let
$\mathfrak{q}\in\mathrm{Ass}(M/N)$. Then
${\mathfrak{p}}^{r}\subseteq\mathfrak{q}$ since ${\mathfrak{p}}^{r}M\subseteq
N$. Therefore ${\mathfrak{p}}\subseteq\mathfrak{q}$. Now $\mathfrak{q}=(N:z)$
for some $z\in M$, $z\notin N$, that is,
$({\mathfrak{p}}^{r}M:z)\subseteq\mathfrak{p}$. Let $a\in\mathfrak{q}$. Then
$az\in N$, which gives $({\mathfrak{p}}^{r}M:az)\not\subseteq\mathfrak{p}$.
Let $b\in R\setminus\mathfrak{p}$ such that $baz\in{\mathfrak{p}}^{r}M$, i.e.,
$ba\in({\mathfrak{p}}^{r}M:z)\subseteq\mathfrak{p}$. This implies
$a\in\mathfrak{p}$. Therefore $\mathfrak{q}\subseteq\mathfrak{p}$. Hence
$\mathrm{Ass}(M/N)=\\{\mathfrak{p}\\}$.

If $N=M_{0}\overset{{\mathfrak{p}}_{1}}{\subset}M_{1}\subset\cdots\subset
M_{k-1}\overset{{\mathfrak{p}}_{k}}{\subset}M_{k}=M$ is an RPE filtration of
$M$ over $N$, then
$\\{{\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{k}\\}=\mathrm{Ass}(M/N)=\\{\mathfrak{p}\\}$.
So ${\mathcal{P}}_{M}(N)={\mathfrak{p}}^{k}$. Suppose $k<r$. Then
${\mathfrak{p}}^{r-1}\subseteq{\mathfrak{p}}^{k}$, which implies
${\mathfrak{p}}^{r-1}M\subseteq{\mathfrak{p}}^{k}M\subseteq N$. So, for every
$x\in{\mathfrak{p}}^{r-1}M$,
$({\mathfrak{p}}^{r}M:x)\not\subseteq\mathfrak{p}$. But
$\mathfrak{p}\in\mathrm{Ass}({{\mathfrak{p}}^{r-1}M}/{{\mathfrak{p}}^{r}M})$
implies $\mathfrak{p}=({\mathfrak{p}}^{r}M:x)$ for some
$x\in{\mathfrak{p}}^{r-1}M$, a contradiction. Therefore, $k\geq r$, and this
implies $M_{r}\subseteq M_{k}=M$. By Lemma 1.3, $M_{r}=\\{x\in
M\mid{\mathfrak{p}}^{r}x\subseteq N\\}$. For any $x\in M$,
${\mathfrak{p}}^{r}x\subseteq{\mathfrak{p}}^{r}M\subseteq N$. Therefore
$M_{r}=M$. So,
$N\overset{\mathfrak{p}}{\subset}M_{1}\overset{\mathfrak{p}}{\subset}\cdots\overset{\mathfrak{p}}{\subset}M_{r}=M$
is an RPE filtration of $M$ over $N$, and hence
${\mathcal{P}}_{M}(N)={\mathfrak{p}}^{r}$. ∎

In Example 1.4,
$\mathfrak{p}\notin\mathrm{Ass}({\mathfrak{p}}/{\mathfrak{p}}^{2})=\\{(\overline{x},\overline{y},\overline{z})\\}$.
So $\mathfrak{p}$ need not be an element of
$\mathrm{Ass}({{\mathfrak{p}}^{r-1}M}/{{\mathfrak{p}}^{r}M})$ even if
${{\mathfrak{p}}^{r}M}\subsetneq{{\mathfrak{p}}^{r-1}M}$.

######  Theorem 2.5.

Let $M$ be an $R$-module, ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ be
distinct prime ideals in $R$ ordered as
${\mathfrak{p}}_{i}\not\subset{\mathfrak{p}}_{j}$ for $i<j$, and
$r_{1},\dots,r_{n}$ be positive integers. If
${\mathfrak{p}}_{i}\in\mathrm{Supp}({{\mathfrak{p}}_{i}}^{r_{i}-1}{{\mathfrak{p}}_{i+1}}^{r_{i+1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)$
for $i=1,\dots,n$, then there exists a submodule $N$ in $M$ such that
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}$.

###### Proof.

We prove by induction on $n$. If $n=1$,
${\mathfrak{p}}_{1}\in\mathrm{Supp}({{\mathfrak{p}}_{1}}^{r_{1}-1}M)$ and by
Proposition 2.2,
${\mathfrak{p}}_{1}\in\mathrm{Ass}({{{\mathfrak{p}}_{1}}^{{r_{1}}-1}M}/{{{\mathfrak{p}}_{1}}^{r_{1}}M})$.
Then by Proposition 2.4, there exists a submodule $N$ in $M$ with
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}^{r_{1}}$. Now let $n>1$, and assume
that the result is true for $n-1$ prime ideals. Then there exists a submodule
$L$ in $M$ with
${\mathcal{P}}_{M}(L)={{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}$.
That is, we have an RPE filtration

$L\overset{{\mathfrak{p}}_{2}}{\subset}L^{(2)}_{1}\overset{{\mathfrak{p}}_{2}}{\subset}\cdots\overset{{\mathfrak{p}}_{2}}{\subset}L^{(2)}_{r_{2}}\overset{{\mathfrak{p}}_{3}}{\subset}L^{(3)}_{1}\overset{{\mathfrak{p}}_{3}}{\subset}\cdots\subset
L^{(n)}_{r_{n}-1}\overset{{\mathfrak{p}}_{n}}{\subset}L^{(n)}_{r_{n}}=M.$
(2.1)

Then ${{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M\subseteq
L$.

So, we have
$\mathrm{Ann}({{\mathfrak{p}}_{1}}^{r_{1}-1}L)\subseteq\mathrm{Ann}({{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)\subseteq{\mathfrak{p}}_{1}$
since
${\mathfrak{p}}_{1}\in\mathrm{Supp}({{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)$.
That is,
${\mathfrak{p}}_{1}\in\mathrm{Supp}({{\mathfrak{p}}_{1}}^{r_{1}-1}L)$, and by
Proposition 2.2,
${\mathfrak{p}}_{1}\in\mathrm{Ass}({{\mathfrak{p}}_{1}}^{r_{1}-1}L/{{\mathfrak{p}}_{1}}^{r_{1}}L)$.
Then by Proposition 2.4, there exists a submodule $N$ in $L$ such that
${\mathcal{P}}_{L}(N)={{\mathfrak{p}}_{1}}^{r_{1}}$. That is, we have the RPE
filtration

$N\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{2}\subset\cdots\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{r_{1}}=L.$
(2.2)

Next, we show that

$N=L^{(1)}_{0}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{2}\subset\cdots\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{r_{1}}=L\overset{{\mathfrak{p}}_{2}}{\subset}L^{(2)}_{1}\overset{{\mathfrak{p}}_{2}}{\subset}\cdots\\\
\overset{{\mathfrak{p}}_{2}}{\subset}L^{(2)}_{r_{2}}\overset{{\mathfrak{p}}_{3}}{\subset}L^{(3)}_{1}\subset\cdots\overset{{\mathfrak{p}}_{n-1}}{\subset}L^{(n-1)}_{r_{n-1}}\overset{{\mathfrak{p}}_{n}}{\subset}L^{(n)}_{1}\overset{{\mathfrak{p}}_{n}}{\subset}\cdots\overset{{\mathfrak{p}}_{n}}{\subset}L^{(n)}_{r_{n}}=M$
(2.3)

is an RPE filtration of $M$ over $N$, which would imply that
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}$.
Since the filtration (2.1) is already an RPE filtration, it is enough to show
that $L^{(1)}_{j-1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{j}$ is a
regular prime extension in $M$ for $1\leq j\leq r_{1}$.

From (2.2) we have that $L^{(1)}_{j-1}\subset L^{(1)}_{j}$ is a
${\mathfrak{p}}_{1}$-prime extension for every $1\leq j\leq r_{1}$. Suppose
$L^{(1)}_{j-1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{j}$ is not a
maximal ${\mathfrak{p}}_{1}$-prime extension in $M$ for some $j$. Then there
exists a submodule $K\supset L^{(1)}_{j}$ such that
$L^{(1)}_{j-1}\overset{{\mathfrak{p}}_{1}}{\subset}K$ is a
${\mathfrak{p}}_{1}$-prime extension in $M$. Since
$L^{(1)}_{j-1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{j}$ is a maximal
${\mathfrak{p}}_{1}$-prime extension in $L$, $K\not\subseteq L$. Let $x\in
K\setminus L$. For $2\leq i\leq n$, since
${\mathfrak{p}}_{1}\not\subseteq{\mathfrak{p}}_{i}$, there exists
$p_{i}\in{\mathfrak{p}}_{1}\setminus{\mathfrak{p}}_{i}$. Then $p_{i}x\in
L^{(1)}_{j-1}$. Since $L^{(1)}_{j-1}\subset L$, from (2.1) we get that
$p_{i}x\in L^{(i)}_{k}$ for every $2\leq i\leq n$, $1\leq k\leq r_{i}$.

Since $p_{n}x\in L^{(n)}_{r_{n}-1}$, $L^{(n)}_{r_{n}-1}\subset M$ is a
${{\mathfrak{p}}_{n}}$-prime extension, and $p_{n}\notin{\mathfrak{p}}_{n}$,
we have $x\in L^{(n)}_{r_{n}-1}$. Then $p_{n}x\in L^{(n)}_{r_{n}-2}$ and
$L^{(n)}_{r_{n}-2}\overset{{\mathfrak{p}}_{n}}{\subset}L^{(n)}_{r_{n}-1}$ is a
prime extension implies $x\in L^{(n)}_{r_{n}-2}$. Repeating this argument
$r_{n}-3$ times, we get $x\in L^{(n-1)}_{r_{n-1}}$.

Replacing $M$ by $L^{(n-1)}_{r_{n-1}}$ and $p_{n}$ by $p_{n-1}$ in the
previous paragraph, we get $x\in L^{(n-2)}_{r_{n-2}}$. Continuing this
process, finally we get $x\in L^{(1)}_{r_{1}}=L$, a contradiction. Therefore,
$L^{(1)}_{j-1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{j}$ is a maximal
prime extension in $M$ for every $1\leq j\leq r_{1}$, and hence (2.3) is an
MPE filtration of $M$ over $N$.

So, for $1\leq j\leq r_{1}$, we get
$\mathrm{Ass}(M/L^{(1)}_{j-1})=\\{{\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}\\}$
and since ${\mathfrak{p}}_{1}\not\subset{\mathfrak{p}}_{i}$ for every $i>1$,
${\mathfrak{p}}_{1}$ is maximal in $\mathrm{Ass}(M/L^{(1)}_{j-1})$. Therefore
(2.3) is an RPE filtration of $M$ over $N$. Hence
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}$.
∎

The converse of Theorem 2.5 does not hold. For example, if
${\mathfrak{p}}_{2}\subsetneq{\mathfrak{p}}_{1}$ are prime ideals in a ring
$R$ and $M$ is the $R$-module
$\frac{R}{{\mathfrak{p}}_{2}}\oplus\frac{R}{{\mathfrak{p}}_{2}}$, then for its
submodule $N=\frac{{\mathfrak{p}}_{1}}{{\mathfrak{p}}_{2}}\oplus 0$, we have
the RPE filtration

$N=\dfrac{{\mathfrak{p}}_{1}}{{\mathfrak{p}}_{2}}\oplus
0\quad\overset{{\mathfrak{p}}_{1}}{\subset}\quad\dfrac{R}{{\mathfrak{p}}_{2}}\oplus
0\quad\overset{{\mathfrak{p}}_{2}}{\subset}\quad\dfrac{R}{{\mathfrak{p}}_{2}}\oplus\dfrac{R}{{\mathfrak{p}}_{2}}=M$

of $M$ over $N$. So we have
${\mathcal{P}}_{M}(N)={\mathfrak{p}}_{1}{\mathfrak{p}}_{2}$. But
${\mathfrak{p}}_{2}M=0$. Therefore
${\mathfrak{p}}_{1}\notin\mathrm{Supp}({\mathfrak{p}}_{2}M)$.

Next, we show that if we assume further that
${\mathfrak{p}}_{i}\not\subset{\mathfrak{p}}_{j}$ for $i\neq j$, the converse
of Theorem 2.5 holds. We need the following lemma.

######  Lemma 2.6.

[2, Lemma 2.8] If $N\overset{\mathfrak{p}}{\subset}K$ is a regular
$\mathfrak{p}$-prime extension in $M$, then for any submodule $L$ of $M$,
$N\cap L\overset{\mathfrak{p}}{\subset}K\cap L$ is a regular
$\mathfrak{p}$-prime extension in $L$ when $N\cap L\neq K\cap L$.

######  Theorem 2.7.

Let $N$ be a submodule of $M$ with
${\mathcal{P}}_{M}(N)={{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}$,
where ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ are distinct prime ideals
in $R$ and $r_{1},\dots,r_{n}$ are positive integers. If all the prime ideals
in $\mathrm{Ass}(M/N)$ are minimal, then
${\mathfrak{p}}_{i}\in\mathrm{Supp}({{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{i-1}}^{r_{i-1}}{{\mathfrak{p}}_{i}}^{r_{i}-1}{{\mathfrak{p}}_{i+1}}^{r_{i+1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)$
for $i=1,\dots,n$.

###### Proof.

Since ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ are minimal, for every $i$
we can reorder ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ such that
${\mathfrak{p}}_{1}={\mathfrak{p}}_{i}$ and by Lemma 1.1, we have an RPE
filtration

$N\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{2}\subset\cdots\subset
L^{(1)}_{r_{1}-1}\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{r_{1}}\overset{{\mathfrak{p}}_{2}}{\subset}L^{(2)}_{1}\subset\cdots\overset{{\mathfrak{p}}_{n}}{\subset}L^{(n)}_{r_{n}}=M$

of $M$ over $N$. So it is enough to show that
${\mathfrak{p}}_{1}\in\mathrm{Supp}({{\mathfrak{p}}_{1}}^{r_{1}-1}$
${{\mathfrak{p}}_{2}}^{r_{2}}$ $\cdots$ ${{\mathfrak{p}}_{n}}^{r_{n}}M)$.
Clearly
${{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M\subseteq N$
and
${{{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M}\subseteq
L^{(1)}_{1}$.

We claim that
${{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M\not\subseteq
N$. Let $x\in L^{(1)}_{r_{1}}\setminus L^{(1)}_{r_{1}-1}$. Then
${{\mathfrak{p}}_{1}}^{r_{1}-1}x\subseteq L^{(1)}_{1}$ and
${{\mathfrak{p}}_{1}}^{r_{1}-1}x\not\subset N$. So there exists
$b\in{{\mathfrak{p}}_{1}}^{r_{1}-1}$ such that $bx\in L^{(1)}_{1}\setminus N$.
Choose $a_{j}\in{\mathfrak{p}}_{j}\setminus{\mathfrak{p}}_{1}$ for every
$2\leq j\leq n$ and let $a=\prod_{2\leq j\leq n}a_{j}^{r_{j}}$. Then
$bax\in{{{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M}$.
Suppose $bax\in N$. Then, since $bx\in L^{(1)}_{1}\setminus N$ and
$N\overset{{\mathfrak{p}}_{1}}{\subset}L^{(1)}_{1}$ is a
${\mathfrak{p}}_{1}$-prime extension, we get $a\in{\mathfrak{p}}_{1}$, a
contradiction. So $bax\notin N$. Therefore
${{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M\not\subseteq
N$. So we have

$\displaystyle
N\,\cap\;({{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)\;$
$\displaystyle\subsetneq\;{{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M$
$\displaystyle=\;L^{(1)}_{1}\,\cap\;({{{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M}).$

Then by Lemma 2.6,

$N\,\cap\;({{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)\quad\overset{{\mathfrak{p}}_{1}}{\subset}\quad{{{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M}$

is a regular ${\mathfrak{p}}_{1}$-prime extension in
${{{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M}$.
Then by Corollary 2.3,
${\mathfrak{p}}_{1}\in\mathrm{Supp}({{\mathfrak{p}}_{1}}^{r_{1}-1}{{\mathfrak{p}}_{2}}^{r_{2}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)$.
∎

From Theorems 2.5 and 2.7, we get the following corollary.

######  Corollary 2.8.

Let ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ be distinct prime ideals in
$R$ with ${\mathfrak{p}}_{i}\not\subset{\mathfrak{p}}_{j}$ for $i\neq j$, and
$r_{1},\dots,r_{n}$ be positive integers. Then
${{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}$ is the
generalized prime ideal factorization of some submodule of $M$ if and only if
${\mathfrak{p}}_{i}\in\mathrm{Supp}({{\mathfrak{p}}_{1}}^{r_{1}}\cdots{{\mathfrak{p}}_{i-1}}^{r_{i-1}}{{\mathfrak{p}}_{i}}^{r_{i}-1}{{\mathfrak{p}}_{i+1}}^{r_{i+1}}\cdots{{\mathfrak{p}}_{n}}^{r_{n}}M)$
for every $1\leq i\leq n$.

In [5] we have found conditions for
${\mathcal{P}}_{M}({{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}M)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$
[5, Theorem 2.14] and showed that this need not always be true [5, Example
2.5]. Now for an $R$-module $M$ and a product of prime ideals
${\mathfrak{a}}={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$
(${\mathfrak{p}}_{i}$’s not necessarily distinct), we give a necessary and
sufficient condition for ${\mathcal{P}}_{M}({\mathfrak{a}}M)={\mathfrak{a}}$.

######  Theorem 2.9.

Let $M$ be an $R$-module and ${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$ be
prime ideals in $R$, not necessarily distinct, with ${\mathfrak{p}}_{i}$
maximal among $\\{{\mathfrak{p}}_{i},\dots,{\mathfrak{p}}_{n}\\}$ for $1\leq
i\leq n$. Let ${\mathfrak{a}}={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$,
${\mathfrak{a}}_{0}=R$, and
${\mathfrak{a}}_{i}={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{i}}$ for
$i=1,\dots,n-1$. Then ${\mathcal{P}}_{M}({\mathfrak{a}}M)={\mathfrak{a}}$ if
and only if
$\mathrm{Ass}\big{(}\frac{({\mathfrak{a}}M:{\mathfrak{a}}_{i})}{({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})}\big{)}=\\{{\mathfrak{p}}_{i}\\}$
for every $1\leq i\leq n$.

###### Proof.

If
$\mathrm{Ass}\big{(}\frac{({\mathfrak{a}}M:{\mathfrak{a}}_{i})}{({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})}\big{)}=\\{{\mathfrak{p}}_{i}\\}$
for every $1\leq i\leq n$, we show that

${\mathfrak{a}}M\overset{{\mathfrak{p}}_{1}}{\subset}({\mathfrak{a}}M:{\mathfrak{a}}_{1})\overset{{\mathfrak{p}}_{2}}{\subset}({\mathfrak{a}}M:{\mathfrak{a}}_{2})\subset\cdots\subset({\mathfrak{a}}M:{\mathfrak{a}}_{n-1})\overset{{\mathfrak{p}}_{n}}{\subset}({\mathfrak{a}}M:{\mathfrak{a}})=M$
(2.4)

is an RPE filtration.

$\mathrm{Ass}\big{(}\frac{({\mathfrak{a}}M:{\mathfrak{a}}_{i})}{({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})}\big{)}=\\{{\mathfrak{p}}_{i}\\}$
implies that there exists a regular ${\mathfrak{p}}_{i}$-prime extension $K$
of $({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})$ in
$({\mathfrak{a}}M:{\mathfrak{a}}_{i})$. Then
$K=\\{x\in({\mathfrak{a}}M:{\mathfrak{a}}_{i})\mid$
${\mathfrak{p}}_{i}x\subseteq({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})\\}$ by
Lemma 1.3. For every $x\in({\mathfrak{a}}M:{\mathfrak{a}}_{i})$,
${\mathfrak{a}}_{i-1}{\mathfrak{p}}_{i}x={\mathfrak{a}}_{i}x\subseteq{\mathfrak{a}}M$,
that is, ${\mathfrak{p}}_{i}x\subseteq({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})$.
Therefore, $K=({\mathfrak{a}}M:{\mathfrak{a}}_{i})$, and hence
$({\mathfrak{a}}M:{\mathfrak{a}}_{i})$ is the unique regular
${\mathfrak{p}}_{i}$-prime extension of
$({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})$ in
$({\mathfrak{a}}M:{\mathfrak{a}}_{i})$. Suppose it is not maximal in $M$. Then
there exists $x\in M\setminus({\mathfrak{a}}M:{\mathfrak{a}}_{i})$ such that
${\mathfrak{p}}_{i}x\subseteq({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})$, i.e.,
$x\in({\mathfrak{a}}M:{\mathfrak{a}}_{i-1}{\mathfrak{p}}_{i})=({\mathfrak{a}}M:{\mathfrak{a}}_{i})$,
a contradiction. So $({\mathfrak{a}}M:{\mathfrak{a}}_{i})$ is a maximal
${\mathfrak{p}}_{i}$-prime extension of
$({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})$ in $M$ for every $i$. Therefore (2.4)
is an MPE filtration of $M$ over ${\mathfrak{a}}M$. This implies that
$\mathrm{Ass}\big{(}\frac{M}{({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})}\big{)}=\\{{\mathfrak{p}}_{i},\dots,{\mathfrak{p}}_{n}\\}$
for every $1\leq i\leq n$. Since ${\mathfrak{p}}_{i}$ is maximal among
$\\{{\mathfrak{p}}_{i},\dots,{\mathfrak{p}}_{n}\\}$, ${\mathfrak{p}}_{i}$ is
maximal in
$\mathrm{Ass}\big{(}\frac{M}{({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})}\big{)}$.
Therefore (2.4) is an RPE filtration. Hence
${\mathcal{P}}_{M}({\mathfrak{a}}M)={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}={\mathfrak{a}}$.

Conversely, suppose that
${\mathcal{P}}_{M}({\mathfrak{a}}M)={\mathfrak{a}}={{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{n}}$.
Since ${\mathfrak{p}}_{i}$ is maximal among
$\\{{\mathfrak{p}}_{i},\dots,{\mathfrak{p}}_{n}\\}$ for every $1\leq i\leq n$,
we can construct an RPE filtration

${\mathfrak{a}}M=M_{0}\overset{{\mathfrak{p}}_{1}}{\subset}M_{1}\overset{{\mathfrak{p}}_{2}}{\subset}M_{2}\subset\cdots
M_{n-1}\overset{{\mathfrak{p}}_{n}}{\subset}M_{n}=M$

of $M$ over ${\mathfrak{a}}M$. Then by Lemma 1.3, $M_{i}=\\{x\in
M\mid{{\mathfrak{p}}_{1}}\cdots{{\mathfrak{p}}_{i}}x\subseteq{\mathfrak{a}}M\\}$,
i.e., $M_{i}=({\mathfrak{a}}M:{\mathfrak{a}}_{i})$ for every $1\leq i\leq n$.
Then clearly
$\mathrm{Ass}\big{(}\frac{({\mathfrak{a}}M:{\mathfrak{a}}_{i})}{({\mathfrak{a}}M:{\mathfrak{a}}_{i-1})}\big{)}=\mathrm{Ass}\big{(}\frac{M_{i}}{M_{i-1}}\big{)}=\\{{\mathfrak{p}}_{i}\\}$
for every $1\leq i\leq n$. ∎

## References

  * [1] T. Duraivel, S. Mangayarcarassy, and K. Premkumar, _Prime extension filtration of modules_ , Int. J. Pure Appl. Math., 98(2) (2015), 211–220.
  * [2] T. Duraivel, S. Mangayarcarassy, and K. Premkumar, _Prime extension dimension of a module_ , J. Algebra Relat. Top., 6(2) (2018), 97–106.
  * [3] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer-Verlag, 1995.
  * [4] H. Matsumura, Commutative ring theory, Cambridge University Press, 1989.
  * [5] K. R. Thulasi, T. Duraivel, and S. Mangayarcarassy, _Generalized prime ideal factorization of submodules_ , J. Algebra Relat. Top., 9(2) (2021), 121–129.

K. R. Thulasi
Department of Mathematics, Pondicherry University, Pondicherry, India.
<EMAIL_ADDRESS>
T. Duraivel
Department of Mathematics, Pondicherry University, Pondicherry, India.
<EMAIL_ADDRESS>
S. Mangayarcarassy
Department of Mathematics, Puducherry Technological University, Pondicherry,
India.
<EMAIL_ADDRESS>


# Halfway to the Peak: The JWST MIRI 5.6 micron number counts and source
population

Leonid Sajkov  Tufts University, MA  Anna Sajina  Tufts University, MA
Alexandra Pope  UMass Amherst, MA  Stacey Alberts  Steward Observatory,
University of Arizona, AZ  Lee Armus  California Institute of Technology, CA
Duncan Farrah  University of Hawaii, HI  Jamie Lin  Tufts University, MA
Danilo Marchesini  Tufts University, MA  Jed McKinney  UT Austin, TX  Sylvain
Veilleux  University of Maryland, MD  Lin Yan  California Institute of
Technology, CA  Jason Young  UMass Amherst, MA

###### Abstract

We present an analysis of 8 JWST Mid-Infrared Instrument 5.6 micron images
with $5\,\sigma$ depths of $\approx$0.1$\mu$Jy. We detect 2854 sources within
our combined area of 18.4 sq.arcmin – a $>4\times$ increase in source density
over earlier IRAC channel 3 data. We compute the MIRI 5.6 $\mu$m number counts
including an analysis of the field-to-field variation. Relative to earlier
published MIRI 5.6 $\mu$m counts, our counts have a more pronounced knee, at
roughly 2 $\mu$Jy. The location and amplitude of the counts at the knee are
consistent with the Cowley et al. (2018) model predictions, although these
models tend to overpredict the counts below the knee. In areas of overlap, 84
% of the MIRI sources have a counterpart in the COSMOS2020 catalog. These MIRI
sources have redshifts that are mostly in the $z\sim 0.5-2$, with a tail out
to $z\sim 5$. They are predominantly moderate to low stellar masses
($10^{8}-10^{10}$M⊙) main sequence star-forming galaxies suggesting that with
$\approx$2 hr exposures, MIRI can reach well below $M^{*}$ at cosmic noon and
reach higher mass systems out to $z\sim 5$. Nearly 70% of the COSMOS2020
sources in areas of overlap now have a data point at 5.6 $\mu$m (rest-frame
near-IR at cosmic noon) which allows for more accurate stellar population
parameter estimates. Finally, we discover 31 MIRI-bright sources not in
COSMOS2020. A cross-match with IRAC channel 1 suggests that 10-20 % of these
are likely lower mass (M${}_{*}\approx 10^{9}$M⊙), $z\sim 1$ dusty galaxies.
The rest (80–90%) are consistent with more massive, but still very dusty
galaxies at $z>3$.

revtex4-1.clsrevtex4-2.cls

##  1 Introduction

Extragalactic astronomy aims to study how galaxies form and evolve across
time. To do so, we build multiwavelength galaxy surveys which allow us to
compare the data with spectral energy distribution (SED) models to infer
redshifts and stellar population parameters (e.g. Brammer et al., 2008; Weaver
et al., 2022a; Wang et al., 2024). Mid-IR data are particularly critical to
this effort since they probe the rest-frame near-IR in the cosmic noon epoch
and earlier which allows for much more accurate stellar masses and star
formation rates (SFRs) (e.g. Elsner et al., 2008; Stefanon et al., 2015;
Muzzin2009l; Martis et al., 2023; La Torre et al., 2024). Indeed, over the
past two decades Spitzer IRAC data – especially its first two channels,
available even in the warm mission (Lacy et al., 2021; Annunziatella et al.,
2023) have been a critical component of these multiwavelength surveys. With
IRAC channel 1 and 2, we reached stellar masses of $\approx 10^{9.5}$M$\odot$
at cosmic noon, i.e. well below the knee of the SMF (Elsner et al., 2008;
Madau & Dickinson, 2014), but only detected the most massive galaxies
($log(M_{*})>11$) at $z>4$ (e.g. Stefanon et al., 2015). The IRAC channel 3,
at 5.8 $\mu$m, which critically covers the rest-frame of the stellar 1.6
$\mu$m bump at $z\sim 2-3$ was relatively underutilized because it was not
available during the extended warm mission and was the least sensitive IRAC
channel even during the cold mission.

The JWST (Gardner et al., 2006), which launched on December 25, 2021, is
already revolutionizing infrared astronomy with its unprecedented performance
(Rigby et al., 2023). In particular, the Mid-InfraRed Instrument (MIRI; Rieke
et al., 2015) allows for significantly greater sensitivity and angular
resolution relative to the Spitzer Space Telescope. Recently published
JWST/MIRI number counts at 7.7 $\mu$m, 10 $\mu$m and 15 $\mu$m bands (Ling et
al., 2022; Wu et al., 2023; Kirkpatrick et al., 2023; Stone et al., 2024) show
dramatic improvement in the depth reached, even with moderate exposure times,
relative to prior measurements from Spitzer and ISO. These counts have already
been used in constraining galaxy and black hole evolution models (Kim et al.,
2024). Number counts at the shortest MIRI wavelength (5.6 $\mu$m) are much
more scarse (Yang et al., 2023; Stone et al., 2024), but they are critical as
this band samples the stellar 1.6 $\mu$m bump at $z\sim 2-3$ and thus is
critical in testing our models of the galaxy population at cosmic noon.

In addition, sampling the rest-frame near-IR at cosmic noon, the 5.6 $\mu$m
band is much less sensitive to the effects of dust obscuration even than
traditionally ‘dust clean’ bands such as the $K$ band, which is rest-frame
$r$-band at the same redshifts. This insensitivity to dust obcuration is
important as in the more than two decades since the discovery of the Cosmic
Infrared Background (e.g. Puget et al., 1996) it has become abundantly clear
that the bulk of star-formation activity at cosmic noon and beyond takes place
in dust obscured galaxies (see Casey et al. 2014 for a review) which in their
extreme are missed in UV/optical surveys (see e.g. Hughes et al., 1998; Sajina
et al., 2006; Pope et al., 2008; Zavala et al., 2021, among many others).
Recently, JWST data have further highlighted this by finding that even deep
HST data miss the reddest/most dust obscured parts of the galaxy population
(e.g. Labbe et al., 2023; Barrufet et al., 2023; Williams et al., 2023).

In this paper, we use deep MIRI 5.6 $\mu$m images to provide a first look at
the properties of the 5.6 $\mu$m number counts and source population based on
images obtained with nearly 2 hr exposure times. This wavelength corresponds
roughly to the rest-frame $H/K$-band at cosmic noon ($z\sim 1-3$), thus
probing primarily stellar mass at these critical redshifts. This paper is
structured as follows. In Section 2 we present the MIRI 5.6 $\mu$m imaging
data and data reduction. In Section 3 we present the source detection and
photometry and the verification of the latter. In Section 4, we show the key
results of our work. These include the 5.6 $\mu$m number counts (Section 4.2);
the redshift distribution and other properties of the MIRI 5.6 $\mu$m sources
with counterparts in COSMOS2020 (Section 4.3); and lastly MIRI 5.6 $\mu$m
sources without counterparts in COSMOS2020 suggesting very red colors (Section
4.4). Throughout we adopt the AB magnitude system (Oke & Gunn, 1974). We adopt
the cosmology model from the COSMOS2020 catalog (Weaver et al., 2022a), which
is a standard $\Lambda$CDM cosmology with H0 = 70 km/s/Mpc, $\Omega_{M,0}$ =
0.3, and $\Omega_{\Lambda,0}$ = 0.7.

##  2 Data

###  2.1 MIRI imaging data

Our data come from JWST/MIRI imaging with the 5.6 $\mu$m filter obtained as
part of the GO1 program “Halfway to the Peak: A Bridge Program To Map Coeval
Star Formation and Supermassive Black Hole Growth” (PIs Pope, Sajina, Yan; PID
1762; Pope et al., 2021). This program observed 8 targets with the MIRI/MRS
instrument: details on the targets and MRS spectra analysis are provided in
Young et al. (2023). Simultaneous with the MIRI/MRS observations, we obtained
MIRI 5.6 $\mu$m imaging in fields adjacent to the MRS targets. This parallel
observing means our imaging fields are effectively blank fields and therefore
ideal for statistical studies such as number counts. In total, we have six
fields in the First Look Survey (FLS) field and two in the COSMOS field. The
FLS fields were observed in July/August 2022 and the COSMOS fields were
observed in December 2022.

Field name | RA | Dec | Exp. time
---|---|---|---
FLS1 | 17:12:28.50 | +58:59:30.12 | 1.85hrs
FLS2 | 17:24:46.96 | +59:15:24.01 | 1.85hrs
FLS3 | 17:21:07.16 | +58:45:39.67 | 1.85hrs
FLS4 | 17:22:49.98 | +59:40:32.35 | 1.85hrs
FLS5 | 17:19:12.87 | +59:28:53.55 | 1.85hrs
FLS6 | 17:13:13.14 | +58:55:22.51 | 1.85hrs
COSMOS1 | 10:01:14.98 | +2:24:36.36 | 1.85hrs
COSMOS2 | 10:01:42.53 | +2:47:26.64 | 1.85hrs
Table 1: The centers of the MIRI imaging fields used in this paper.

We obtain data over the MIRI imager with an FOV of $74\times$113′′ as well as
the smaller MIRI coronograph with a FOV of $24\times$24′′ (as seen in Figure
1). For our eight pointings, this adds up to a total area of just under 19.4
sq.arcmin. The effective area we use is a bit more restricted after masking
out noisy edges and bright stars. This is discussed in Section 3.2.

Figure 1: An example of our MIRI images (for the FLS6 field) highlighting the
number of sources clearly detected within the imager’s FOV of only 2.3
sq.arcmin. The angular resolution is also illustrated in the zoomed-in insets,
both measuring 9$\times$9′′, where we see both resolved morphology as
illustrated by the detected spiral structure (top inset) as well as resolved
closely spaced sources (that would have been blended in the Spitzer IRAC
images, middle inset). The 2′′ scale bars are comparable to the IRAC channel 3
FWHM which is 1.88′′.

Figure 2: Example images produced with the standard pipeline, and following
two different background removal methods, as described in Section 2.2. The
image median (in counts/second/pixel) are given in the bottom-left of each
panel. Left: the FLS6 image from the standard pipeline. Middle: the FLS6 image
where we have subtracted a median background constructed from all six FLS
fields, with the sources masked out. Right: the FLS6 image with super
background subtraction (Pérez-González et al., 2024; Alberts et al., 2024).

###  2.2 Data reduction

We first obtained our MIRI images from the archive already reduced using the
JWST operations pipeline build 10.1 (science calibration pipeline version
1.12.5; CRDS version 11.17.6, context jwst_1193.pmap)111As of the time of
writing, this was the latest pipeline build that affected MIRI imaging. See
https://jwst-docs.stsci.edu/jwst-science-calibration-pipeline-overview/jwst-
operations-pipeline-build-information. Figure 2 left shows an example of one
of our images. Within these pipeline-reduced images we noted both the presence
of an overall background level (the median pixel value for the FLS images was
1.23, for the COSMOS images it was 1.52). This difference in overall
background level is unsurprising given that FLS is a northern field, whereas
COSMOS is equatorial and as such sees a higher level of Zodiacal light, see
example discussion in Sanders et al. (2007). We also noted some clear striping
especially toward the bottom quarter of each image.

We first attempted a simple background subtraction motivated by the fact that
the stripping patterns were fairly consistent among the images. We thus
constructed a median background image where we used the segmentation map
provided with pipeline-processed images to mask out bright sources in the
fields. To avoid any remaining source signal, we added a 5 pixel “buffer”
around each source within the segmentation maps. Due to the difference in
overall pixel level between the images, we rescaled each source-subtracted
image to a median of one. Figure 2 middle shows an example of our image after
this basic median background subtraction when we only use the FLS images for
constructing the background. It is clear that this method works well in
removing the vast majority of the striping though some residual structure
remains due to second order differences between images. In addition, since we
only have two COSMOS images we could not construct a COSMOS only median using
this method. We attempted a combined median background in FLS and COSMOS;
however, this led to the COSMOS fields (which have less weight in this median
with 2/8 images only) to have higher noise levels. Specifically, using a
single median we found 5 $\sigma$ levels of $\approx$0.095 $\mu$Jy in FLS and
$\approx$0.135 $\mu$Jy in COSMOS. When using only the FLS images for a median
(as shown in Figure 2 middle), we find 5 $\sigma$ levels of $\approx$0.085
$\mu$Jy. While we accounted for the differences in overall background level,
these results clearly indicate that we have some time variable aspects to the
background which make it sub-optimal to construct such median background from
observations about 6 months apart.

To address these issues, we then processed our images through the super-
background subtraction procedure developed by the SMILES (PID 1207; PI G.
Rieke; Lyu et al. (2024)) team (Pérez-González et al., 2024; Alberts et al.,
2024). This background subtraction starts with the stage 2 images and
homogenizes the background across all images in the program taking into
account the time varying behavior of the background which is particularly
prominent at shorter wavelengths as here. The full details are described in
Alberts et al. (2024). Figure 2 right shows the result of this super
background subraction. It is clear that the residual structure in our previous
simple background-subtracted image is now largely gone. In addition, this
method allows for COSMOS only median since it no longer uses the fully
processed images (of which there are only two), but the stage 2 images of
which we have 2$\times$6 dithers which is sufficient for a median background.
As we show in Section 2.4 below, we also no longer find a difference in the
obtained depth in FLS and COSMOS. The depth we find in FLS relative to the
simple background subtraction using the FLS images alone however are
comparable. This suggests that depending on the specific science applications
and data available, a simple procedure as described above may be sufficient,
but it should only be applied to images obtained close in time to avoid time-
varying background effects.

###  2.3 Noise properties

To explore the noise properties of our images post background subtraction, we
constructed a noise map where each pixel value equals the standard deviation
in the image pixel values in a 3$\times$3 pixel grid around that pixel. We
performed this calculation on each image with two masks applied. First, we
masked the pixels belonging to sources by applying the segmentation map. We
then created a mask that removes the image edges and the residual strips, as
we consider them to be sources of additional noise. This second mask is the
same for each image. We perform our noise calculation on the science images
after the segmentation map and noise/stripe mask have been applied. The
resulting noise map for the FLS1 field is shown in Figure 3. We compared our
produced noise map to the one provided with the standard pipeline products
(which is derived from the per pixel coverage). The typical pixel values of
these calculated noise maps are 60-70% of the MAST-provided error maps.

Figure 3 shows a wavy pattern across the whole image. We found this pattern in
all our images, even when varying the number of pixels used in calculating the
standard deviation, and also when running the same procedure before or after
background subtraction. We also found this pattern with both the simple and
super background subtracted images. We did not find this pattern when running
the same procedure on one of the publicly-available CEERS images in the same
filter (Finkelstein et al., 2022). The CEERS images have lower exposure times
(0.82 vs. 1.85 hrs), and slightly smaller pixel scales (0.09′′ vs 0.11′′). At
present, we consider this an unexplained instrumental effect that is
consistent with the description of “tree rings” that have previously been
observed in MIRI imaging222See https://jwst-docs.stsci.edu/jwst-calibration-
pipeline-caveats/jwst-imaging-pipeline-caveats for details on the “tree ring”
patterns.. This pattern is at a very low level, with the amplitude of the
peaks and troughs translating to $\approx$ 0.5 nJy/pixel, as can also be seen
in the scale color bar.

###  2.4 Image depth

We estimate the image depth by placing random apertures in source-free parts
of the background-subtracted images. The standard deviation of the Gaussian
distribution of the resultant empty aperture fluxes is the $1\,\sigma$ depth
of the image. The above method is a simplification of the method used in
Annunziatella et al. (2023), since we have fairly uniform coverage unlike that
in Annunziatella et al. (2023) so noise-weighting is not critical. In Section
3.1 we describe our SExtractor photometry. For our random apertures we use a
diameter equal to the median of the SExtractor Kron radii, where a Kron radius
is a “reduced pseudo-radius” that defines a circular aperture containing
$>90\%$ of the flux of the source333Refer to
https://sextractor.readthedocs.io/en/latest/Photom.html#automatic-aperture-
flux-flux-auto and section 6 of Bertin, E. & Arnouts, S. (1996) for details..
We choose Kron apertures as they are meant to capture most of the light of our
sources, which are typically spatially extended. The median Kron radius in our
catalog is $\sim$0.385′′, so we adopt a diameter of 0.77′′ for our apertures.
For comparison, the FWHM of the MIRI PSF at 5.6 $\mu$m is 0.207′′.

We placed 500 empty apertures per field for a total of 4000 apertures. For
each image, we then fitted a Gaussian to the resulting histogram of aperture
flux. We found that the standard deviations of these fitted Gaussians were
consistent between all the images, translating to a $5\,\sigma$ depth of 0.1
$\mu$Jy.

As a cross-check, the median flux uncertainties for our sources based on the
SExtractor photometry (Section 3.1) translate to a 5 $\sigma$ uncertainty of
0.081 $\mu$Jy. This is reasonably consistent with our image depth analysis
above. Note that the random apertures method is sensitive to the adopted
aperture sizes and slightly smaller apertures would bring the two values into
even closer agreement. As an additional cross-check, the image depths quoted
by CEERS (Yang et al., 2023), rescaled to our exposure time (since depth
scales as $\sqrt{t}$) translate to $\approx$0.08 $\mu$Jy for our images. To be
conservative, we adopt $\approx$0.1 $\mu$Jy as our nominal 5 $\sigma$ image
depth.

Figure 3: Noise map for the FLS1 field. Produced by computing the standard
deviation of the FLS1 image inside a $3\times 3$ pixel square centered on each
pixel. The science image used in producing the noise map was first multiplied
by the field’s segmentation map, removing the sources. We then applied a mask,
shown here in orange, that removes edges and residual stripes as additional
sources of noise. After both steps, we produced the noise map.

###  2.5 Photometric zero-point

Pre-launch estimates of the MIRI imager’s performance already predicted that
its point spread function (PSF) has a ‘cruciform’ artifact that draws
luminosity from the core into the wings of the Gaussian. In-flight
measurements have shown that this artifact is prominent at shorter wavelengths
like ours (Rigby et al., 2023), but is negligible at $>$10 $\mu$m (Gáspár et
al., 2021). The extent of this effect was only noted once in-flight
assessments had been made. Since our data were obtained very early after the
start of science operations, the FITS header keyword PHOTUJA2 provided for
converting from the native units to physical photometry units was incorrect.
The fiducial keyword translated to a photometric zero-point of 26.121 mag
which resulted in us seeing a significant systematic offset in the photometry
measured by MIRI F560W as compared to the Spitzer IRAC Ch3 data available for
our fields.

This issue was also noted by the CEERS team and led them to calculate a
photometric zero-point for the 5.6 $\mu$m images of 25.701 in AB magnitudes
(Papovich et al., 2023). However, the CEERS images have a pixel scale of
0.09$\arcsec$ (Papovich et al., 2023) whereas ours is 0.11$\arcsec$. We re-
scale the CEERS calculated zero-point as in
$zp=25.701-5\log_{10}(0.11/0.09)=25.265$. This is the value we adopt in this
paper which leads to much better agreement between the MIRI 5.6 $\mu$m and
IRAC channel 3 photometry for cross-matched sources (see Section 3.3).

##  3 Data analysis

###  3.1 SExtractor photometry

We used Source Extractor (Bertin, E. & Arnouts, S., 1996) version 2.28.0 for
source detection and photometry. Table 2 lists our SExtractor parameter
settings. The SEEING$\\_$FWHM keyword adopted here corresponds to the FWHM of
the in-flight measured MIRI 5.6 $\mu$m PSF (see section 4.1 for details). As
our fiducial flux measurements, we adopt values produced with Kron aperture
photometry, labeled by the FLUX_AUTO keyword by SExtractor. The photometric
uncertainties are natively provided by SE and take into account the variance
of the pixel values inside the aperture, the background value, and the
effective detector gain.

Extraction |
---|---
DETECT_MINAREA | 9.0
DETECT_THRESH | 1.5
ANALYSIS_THRESH | 0.4
FILTER_NAME | gauss_3.0_5x5.conv
Photometry |
PHOT_AUTOPARAMS | 2.5, 3.5
PHOT_FLUXFRAC | 0.5
MAG_ZEROPOINT | 25.265
GAIN | 36630.528
PIXEL_SCALE | 0.11091469
Background |
BACK_TYPE | AUTO
BACK_SIZE | 16
BACK_FILTERSIZE | 3
BACKPHOTO_TYPE | LOCAL
Deblending |
DEBLEND_NTHRESH | 32
DEBLEND_MINCONT | 0.003
Star/Galaxy Separation |
SEEING_FWHM | 0.207
Table 2: SExtractor parameter settings for the photometry.

###  3.2 Masking

We further clean the raw SExtractor catalogs by masking out bright stars as
well as any particularly noisy parts of the images. To do so we created masks
for each image. These masks include two components. One excludes obviously
diffracted sources. The other avoids all edges (assuming an edge width of 5
pixels), where edges include the blocked regions of the Lyot coronograph. We
also manually remove the noisy residual stripes near the bottom of the images
(see right-hand panel in Figure 2). Figure 3 shows as an example the mask for
the FLS1 field.

We also apply a quality cut on our catalogs, only keeping sources with a
signal-to-noise ratio greater than 5. The combination of the masking and the
quality cut reduces the raw SExtractor output of 4,089 sources to a total of
2,854 “reliable” sources (“reliable” meaning falling outside our defined masks
and having SNR$\geq 5$) among all our eight fields. Per field, we detect the
most in field FLS3 (409 sources) and the least in field FLS5 (310 sources).

Figure 4: Comparison of the position centroids of the 582 matched sources
between MIRI and the COSMOS2020. The histograms (top and right) represent the
offsets in both fields combined, with the mean absolute deviation (MAD,
calculated as the mean of $|\Delta\mathrm{RA}|$ and $|\Delta\mathrm{Dec}|$,
respectively) labeled.

###  3.3 Astrometric & photometric verification

In order to perform astrometric and photometric verifications, we used the
SExtractor detected sources in our two COSMOS fields and cross-matched them
with the COSMOS2020 CLASSIC catalog (Weaver et al., 2022a). The COSMOS2020
source detection is based on a weighted combined $izYJHK$ image (Weaver et
al., 2022a). We used a 1′′ matching radius and found a total of 582 COSMOS2020
cross-matches, 175 of which also have IRAC channel 3 detections. Overall, our
MIRI data increase the fraction of COSMOS2020 with 5.6/5.8 $\mu$m data by
roughly 3 times. Note that our MIRI fields are within the deep part of the
COSMOS2020 catalog and not near the edges and the COSMOS2020 positions rely on
Gaia astrometric solutions (Weaver et al., 2022a). Figure 4 shows the
comparison between the MIRI and COSMOS2020 source positions of the cross-
matched sources. The offsets we find are small and well within the MIRI PSF at
this wavelength. Therefore we do not explicitly apply astrometric corrections.

Figure 5 shows the comparison between the MIRI 5.6 $\mu$m and the available
IRAC channel 3 fluxes. As described in section 3.1, we use Kron aperture
photometry for the 5.6 $\mu$m fluxes. For the IRAC fluxes, we used the
aperture corrected 2′′ aperture photometry from Weaver et al. (2022b). We
overlay the 1:1 line. Overall, we find good agreement, although we have a
median offset of 0.22 magnitudes.

Figure 5: Photometry cross-check for the matched sources with available IRAC
channel 3 measurements applying a $\mathrm{SNR}>1$ cut on the IRAC ch3 data,
leaving us with 119 sources. The dashed diagonal line is the 1:1 line. The
vertical dashed line represents the $3\sigma$ IRAC channel 3 limit (Weaver et
al., 2022a). The quoted median offset is computed for the SNR$>3$ sources.

###  3.4 Estimating fraction of fake sources

The cross-match with COSMOS2020 also gives us an upper limit on the fraction
of the SExtractor detected sources that might be fake. Such sources are
expected due to some of the residual artifacts in the images, that may have
been missed in the masks described in Section 3.2. Overall we found COSMOS2020
counterparts for 582 of the 690 MIRI sources in our two COSMOS fields (84 %).
The 108 unmatched sources ($16\,\%$), represent an upper limit on the fake
source fraction since it is expected that we have some red MIRI sources that
are not detected in COSMOS2020. Figure 6 shows the 5.6 $\mu$m flux
distributions of the COSMOS2020 undetected sources vs. the matched sources.
Unsurprisingly, the unmatched sources are generally fainter than the overall
MIRI source population. Indeed, visual examination of the 108 unmatched
sources showed only 31 are unambiguous MIRI detections. Removing these sources
which are clean MIRI detections despite being unmatched in COSMOS2020 leaves
us with 77 likely fake sources ($\sim 10\%$ of the total). Many of these
apparently fake sources are driven by single bright pixels that were missed in
the standard pipeline reduction, others appear to be residuals of the striping
pattern seen in Figure 2. In Figure 6 we show separately the flux
distributions of the unmatched sources judged to be real vs. those that are
potentially fake. Note that the higher Section 4.4 explores in more detail the
nature of the 31 reliable MIRI sources that are unmatched in COSMOS2020.

Figure 6: A comparison between the flux distributions of our MIRI sources that
are matched to COSMOS2020 sources vs. those that are unmatched.

##  4 Results

###  4.1 Source completeness

In order to estimate the source detection completeness at different flux
density levels, we take the usual Monte Carlo approach by injecting fake
sources at different flux levels and noting the fraction of them that are
recovered with the same SExtractor procedure as applied to real sources (e.g.
Takagi et al., 2012). We use the code described in Shipley et al. (2018) for
this step.

For the injected fake sources, we need a model of their on-the-sky
distribution. Simply adopting the MIRI PSF, is not appropriate since the vast
majority of our sources are spatially resolved. To create a more realistic
source model, we stack 500 isolated and visually compact sources to construct
a source model stamp. Figure 7 shows this stamp which measures 16 pixels
$\times$ 16 pixels ($\approx$1.8${}^{\prime\prime}\times$1.8′′). This source
model stamp has a mean (along x- and y-axis) full width at half-maximum (FWHM)
of 0.357′′ which, as expected, is larger than the MIRI 5.6 $\mu$m PSF of
0.207′′, as measured during JWST commissioning444For details, refer to JWST
user documentation at: https://jwst-docs.stsci.edu/jwst-mid-infrared-
instrument/miri-performance/miri-point-spread-functions..

Figure 7: The $16\times 16$ pixel source model stamp used in the Monte Carlo
simulations to determine source completeness. The FWHM of the best-fit
Gaussian is 0.341′′ in the x-direction and 0.330′′ in the y-direction.

Completeness estimation starts with a segmentation map produced by SExtractor
where all pixels with zero value correspond to no source present. We
additionally set all pixel values to zero if they were masked, as described in
Section 3.2, as well as setting to zero all sources consisting of 25 or fewer
contiguous pixels. This second step was done in order to provide more sites
suitable for injection, as our procedure avoids injecting in sites flagged by
the segmentation map as having sources, though we deem this unnecessary for
smaller sources. We then proceeded to inject fake sources across a range of
flux densities ranging between 0.01 $\mu$Jy and 0.25 $\mu$Jy in 25 total
increments. Sources are injected only on valid pixels not removed in the above
steps. At each flux level, we injected 500 sources in the image. The sources
are injected by using the source model scaled to the desired flux density. We
then processed the newly-created images with the same SExtractor procedure
used for the science images themselves. The ratio of recovered to injected
sources at each flux density level constitutes the completeness at that level.
This procedure was done 10 times per flux level per image, totaling 80 runs.
Figure 8 shows the measured completeness curves for each run as well as the
adopted completeness curve, which is the median between the 80 runs. As
expected we find that the completeness is essentially 100% above our estimated
5 $\sigma$ image depth and drops rapidly below that. The 50% completeness is
at $\approx$0.07 $\mu$Jy.

Figure 8: Source completeness for our images. The $x$-axis is trimmed to show
detail. A sample of the computed completeness curves is faintly overlaid to
show spread. The thin dotted line shows the $5\sigma$ depths of the images as
determined by the empty apertures method, detailed in section 2.4. Figure 9:
Left The integral number counts of our individual 8 fields. For comparison we
also show the combined counts with Poisson errors. We find that the typical
field-to-field variation is a factor of roughly two – well about the simple
Poisson errors on the combined counts. We also overlay the $5\,\sigma$ limit
as a vertical dashed line.
Right Our integral number counts for our eight fields (orange symbols), with
the Poisson and field-to-field variation uncertainties added in quadrature. We
overlay for comparison both the published CEERS counts (Yang et al., 2023,
blue crosses, include only Poisson error) and the SMILES counts (Stone et al.,
2024, green circles, include only Poisson error), as well as model predictions
MIRI F560W counts from Cowley et al. (2018, black curves) where the dot-dash
curve is their fiducial model and the upper (dashed) one is the same but with
evolving supernovae feedback (“EvolFB”). We also overlay the $5\,\sigma$ limit
as a vertical dashed line.

###  4.2 Number counts

We calculate the MIRI 5.6 $\mu$m integral number counts by first counting the
number of objects in each flux bin $i$ as $dN/dS_{i}$. The source completeness
for each flux bin is $C_{i}$ and is as shown in Figure 8. The integral counts
are given by:

$N(>S)=\sum_{S_{i}=S}^{\infty}\frac{1}{A_{\mathrm{eff},i}}\frac{1}{C_{i}}\frac{dN}{dS_{i}}$
(1)

We calculate the effective area, $A_{\mathrm{eff}}$, as the inverse of the
Boolean mask used for source detection (Section 3.2) times the single pixel
area. For all 8 fields combined, this adds up to 18.4 sq.arcmin. Note that our
coverage is fairly uniform within these masks therefore we assume the
effective area does not change with flux level. The calculated integral counts
are given in Table 3 along with their Poisson uncertainties. To calculate the
raw counts, i.e. the ones not corrected for source comleteness, we simply
adopt $C_{i}\equiv 1$ in all flux bins in Equation 1. When computing our
number counts, we use less stringent quality cut on our catalog, counting all
sources with an SNR $\geq 1$. In Figures 9 we overlay the $5\sigma$ limit of
$\approx$0.1 $\mu$Jy on our counts figures as reference.

Figure 9 left shows both our combined MIRI 5.6 $\mu$m integral number counts,
with Poisson errors, and the counts estimated for each of our 8 fields
individually. It is obvious that the field-to-field variation is significantly
greater than the Poisson uncertainties on the combined counts. The shape of
the counts is fairly consistent with nearly all fields showing a knee at
around 6-8 $\mu$Jy.

Figure 9 right shows the combined integral MIRI 5.6 $\mu$m number counts where
the uncertainties on the points represent the total uncertainties. The later
are estimated by adding in quadrature the Poisson uncertainties with a field-
to-field variation uncertainty which is taken to be half of the total spread
between the individual field counts at each flux level. The combined counts
and their Poisson and total uncertainties are given in Table 3.

For comparison, in Figure 9 we overlay the CEERS 5.6 $\mu$m counts from (Yang
et al., 2023). The data for the CEERS number counts come from four nearby
fields (separated by $<$15′) with a total area of $\sim 9.5\,$sq.arcmin. The
median $5\sigma$ depth for the CEERS images at 5.6 $\mu$m is 0.138 $\mu$Jy
(Yang et al., 2023). We note that our counts tend to be higher than the CEERS
ones, but this disagreement is only at $\approx$1 $\sigma$ level for most flux
bins considering our total uncertainties that account for field-to-field
variation. Note that the uncertainties plotted on the CEERS counts are purely
Poisson and do not include field-to-field uncertainties. The gap between our
counts and their only becomes significant at the lowest flux bins, below
$\approx$0.3 $\mu$Jy; the widest part of the gap is $\sim 2.3\sigma$. We also
overlay the just released MIRI 5.6 $\mu$m counts from the SMILES team (Stone
et al., 2024). They are in good agreement with our counts at the bright end,
but closer to CEERS at the faint end, though again within our 1 $\sigma$
uncertainties. We note that both CEERS and SMILES use combined F560W+F770W
images for source detection, unlike this analysis which is done entirely with
MIRI 5.6 $\mu$m images. To test the potential effect of differences in
photometry, we ran our SExtractor setup on the public background-subtracted
CEERS images and compared with their photometric catalog. We find excellent
agreement, with the largest discrepancies per flux bin (e.g. for $<$ 1 $\mu$Jy
fluxes) being that we find roughly 10% more sources. This is consistent with
our estimated fake source fraction in Section 3.4. Such fake sources are less
likely in the CEERS photometric catalog due to their including the F770W as
well for source detection. This accounts for some of the offset we see at
fainter fluxes, but $\approx$10% likely fake source fraction is far from
sufficient to explain the gap which is $\approx$2 $\times$ at its maximum. Our
Figure 9 left suggests this is primarily due to field-to-field variation
instead.

In Figure 9 right we also overlay the model counts from Cowley et al. (2018).
Their predictions are based on the GALFORM semi-analytic model for galaxy
formation within a $\Lambda$CDM framework (Lacey et al., 2016). The
predictions based on the core galaxy formation model from Lacey et al. (2016)
are referred to as the ”Baseline model”. The model that includes supernovae
feedback whose strength evolves with redshift is called the “EvolFB
model”555We obtained the Cowley et al. (2018) model data from
http://icc.dur.ac.uk/data/.. The GALFORM model assumes the Maraston (2005)
stellar population models and the dust radiative transfer comes from the
GRASIL prescription of Silva et al. (1998). Our counts, in the brighter
regime, are reasonably consistent with these model predictions with a slight
preference for their evolving feedback model, although the difference between
their models is well within our uncertainaties (including the field-to-field
variation). In agreement with the CEERS and SMILES results, we find that the
Cowley et al. (2018) models overestimate the counts for sources below
$\approx$3 $\mu$Jy.

Table 3: The MIRI 5.6 $\mu$m number counts

log($S_{560W}$) | Raw $N(>S)$ | Corrected $N(>S)$ | Poisson error | Total error lower-upper$\dagger$$\dagger$footnotemark:
---|---|---|---|---
[$\mu$Jy] | [10-5deg${}^{-}2$] | [10-5deg${}^{-}2$] | [10-5deg${}^{-}2$] | [10-5deg${}^{-}2$]
0.05 | 5.49 | 6.82 | 0.10 | 1.37-1.40
0.07 | 5.40 | 6.01 | 0.10 | 0.96-1.05
0.09 | 5.01 | 5.10 | 0.10 | 0.63-0.60
0.12 | 4.41 | 4.41 | 0.09 | 0.54-0.55
0.16 | 3.79 | 3.79 | 0.09 | 0.58-0.55
0.22 | 3.23 | 3.23 | 0.08 | 0.49-0.53
0.31 | 2.65 | 2.65 | 0.07 | 0.50-0.46
0.42 | 2.25 | 2.25 | 0.07 | 0.58-0.28
0.58 | 1.87 | 1.87 | 0.06 | 0.58-0.30
0.80 | 1.54 | 1.54 | 0.05 | 0.45-0.30
1.10 | 1.32 | 1.32 | 0.05 | 0.38-0.21
1.51 | 1.12 | 1.12 | 0.05 | 0.35-0.23
2.07 | 0.94 | 0.94 | 0.04 | 0.29-0.24
2.85 | 0.80 | 0.80 | 0.04 | 0.28-0.22
3.92 | 0.68 | 0.68 | 0.04 | 0.26-0.14
5.39 | 0.54 | 0.54 | 0.03 | 0.22-0.17
7.41 | 0.43 | 0.43 | 0.03 | 0.22-0.13
10.18 | 0.31 | 0.31 | 0.02 | 0.13-0.06
14.00 | 0.21 | 0.21 | 0.02 | 0.08-0.08
19.25 | 0.13 | 0.13 | 0.02 | 0.07-0.08
26.47 | 0.07 | 0.07 | 0.01 | 0.04-0.07
36.39 | 0.05 | 0.05 | 0.009 | 0.03-0.05
50.04 | 0.02 | 0.02 | 0.006 | 0.0086-0.026
68.80 | 0.0058 | 0.0058 | 0.003 | 0.010-0.025
94.60 | 0.0019 | 0.0019 | 0.002 | 0.014-0.014

††footnotetext: These uncertainties include Poisson as well as field-to-field
variation.

Figure 10: The COSMOS1 field with all COSMOS2020 Classic catalog (Weaver et
al., 2022a) sources overlaid as the green circles. The COSMOS2020 sources
which have IRAC channel 3 (5.8 $\mu$m) detections as shown as green triangles.
The orange diamonds are all our MIRI-detected sources in this field. Note that
we have $>$3$\times$ more MIRI 5.6 $\mu$m detections in the field relative to
the prior Spitzer IRAC 5.8 $\mu$m detections.

Figure 11: The redshift distribution of our COSMOS2020 matched MIRI 5.6 $\mu$m
sources. These are LePhare redshifts from the classic COSMOS2020 catalog.

###  4.3 The MIRI 5.6 $\mu$m sources in the COSMOS2020 catalog

As already discussed in Section 3.3, we cross-matched the sources within our
two COSMOS fields with the COSMOS2020 catalog (Weaver et al., 2022a, b). The
overlap between our MIRI detections, the COSMOS2020 sources and their subset
with IRAC channel 3 detections is illustrated in Figure 10 in the case of the
COSMOS1 field. Note that with these new data more than half of the COSMOS2020
sources now have 5.6 $\mu$m detections. The number of sources with MIRI 5.6
$\mu$m detection in this field is more than 4$\times$ the sources with
previous IRAC channel 3 (5.8 $\mu$m) detections. This is illustrative of the
fact that until JWST, it was difficult to link the optical and mid-IR source
populations. Within our two COSMOS MIRI fields, there are 690 MIRI sources, of
which 582 (84%) are matched to COSMOS2020 catalog sources, using a 1$\arcsec$
matching radius (see Section 3.3).

Figure 11 shows the redshift distribution of the MIRI sources with
counterparts in the COSMOS2020 catalog. We show the LePhare redshifts;
although the distribution is very similar using the EAZY redshifts. In Figure
11, we applied a quality cut of
$0.01<|z_{u68}-z_{l68}|/(1+z_{\mathrm{med}})<0.5$, where $u68$, $l68$, and
$\mathrm{med}$ denote the upper and lower 68% confidence intervals and the
median of the redshift probability density function. Of the 582 matched
sources, 520 passed the quality cut and are presented the figure. This
histogram shows peaks at $z\sim 1$ and $z\sim 2$, i.e. we find many cosmic
noon galaxies.

Up to $z\sim 2.5$, detection is aided by a negative $k$-correction as the
observed 5.6 $\mu$m probes up the SED towards the stellar 1.6 $\mu$m bump,
which explains the sharper drop in sources at higher redshift. However, we do
see a tail out to $z\sim 5$ which corresponds to early massive galaxies.

Figure 12 shows the stellar mass vs. star-formation rate for our
COSMOS2020-matched MIRI sources. We find predominantly star-forming main
sequence galaxies with stellar masses in the $10^{8}-10^{10}$ M⊙ range. We are
probing the typical low-mass star-forming galaxy at cosmic noon, well below
$M^{*}$ ($\approx 10^{11}$M⊙) at cosmic noon (Adams et al., 2021).

###  4.4 The nature of MIRI sources not in COSMOS2020

As discussed in Section 3.4, while the bulk of our sources (84%) have
counterparts in the COSMOS2020 catalog, 108 MIRI are un-matched. We examined
these by eye to remove sources that have likely photometric issues such as
being driven by a single hot pixel or being residuals of the stripes we saw
prior to background subtraction (see Figure 2). After this visual inspection,
we are left with 31 reliable MIRI detections not present in the COSMOS2020
catalog. This translates to a source density of $\sim$7/sq.arcmin.

Figure 12: The stellar mass vs. star formation rate relation for the same
sources presented in Figure 11. The overlaid lines are the main sequence
relations at the indicated redshifts based on Koprowski et al. (2024).

To understand why strong MIRI detected sources may not be in the COSMOS2020
catalog consider that the COSMOS2020 source detection is based on a
$\chi^{2}$-weighted $izYJHK$ image (Weaver et al., 2022a). Figure 13 shows the
5 $\sigma$ limits of the COSMOS2020 catalog in these six bands, as well as the
four IRAC channels. In all cases, we take the limits from the deep parts of
COSMOS since these correspond to the locations of our two COSMOS fields. Note
that our MIRI F560W depth is comparable to the IRAC channel 1 and 2 depths in
this field, while the IRAC channel 3 image is approximately 3 magnitudes
shallower (see figure 13). We overlay two potential models that could explain
non-inclusion in the COSMOS2020 catalog, but detection in MIRI 5.6 $\mu$m. One
is a $z=1$ $M_{star}=10^{9}M\odot$ star-forming galaxy with $A_{V}=3$ (the
“Low-mass dusty” model). The other is a $z=4$ $M_{star}=5\times 10^{10}M\odot$
star-forming galaxy with $A_{V}=3$ (the “High mass dusty” model). Here we use
the CSP models from Maraston (2005), assume an age of 300Myr for both galaxies
and use the Calzetti law (Calzetti et al., 1994) for dust attenuation. The
first model is included because while, previously, low mass galaxies were
typically not considered very dusty, JWST has recently found evidence of a
population of $z<2$ extremely dusty dwarfs (with masses
$10^{7}-10^{9}M_{\odot}$ and up to $A_{V}\sim 5$ , Bisigello et al. 2023; see
also Pope et al. 2023).

Figure 13: We compare the 5 $\sigma$ magnitude limit of our images (in red)
with the $izYJHK$+IRAC 5 $\sigma$ magnitude limits for the COSMOS2020 catalog
(in grey), based on Weaver et al. (2022a). We suggest two potential
explanations for the MIRI detections that are not in the COSMOS catalog:
either a low mass intermediate ($z\sim 1$) redshift galaxy or a galaxy closer
to $M_{*}$ at $z\sim 3$ or beyond with high dust content. See Section 4.4 for
more details on the models.

In Figure 13, our model spectra are chosen to both lie below the limits for
source detection in COSMOS2020 but also to have MIRI 5.6 $\mu$m magnitudes on
par with those of the sources shown in Figure 14. Both models satisfy these
criteria, but they are clearly distinguishable by the IRAC channel 1
brightness. Therefore in Figure 14 we overlay our MIRI images of these 31
sources with the 2 and 3 $\sigma$ contours from the COSMOS IRAC channel 1
image. We used the IRAC channel 1 mosaic combining all available COSMOS data,
as produced by Annunziatella et al. (2023). We find that 3-7 of the sources
have emission in IRAC channel 1. The range given is to distinguish clean
detections from ones affected by blending in the IRAC image. The three clean
IRAC detections in particular (ID 1-16, 1-107, 1-110) are all consistent with
the low-mass dusty model. However, the bulk of our 31 sources are undetected
in these IRAC images suggesting they are higher-mass and higher redshift dusty
sources. Further study of these 31 sources, folding on the new COSMOS-Web data
(Casey et al., 2023), as well as all other available data in COSMOS is
reserved for a separate paper.

Figure 14: $3.3\times$3.3′′ cutouts of the 31 sources detected in MIRI 5.6
$\mu$m but are not present in the COSMOS 2020 Classic catalog. The red
contours represent the 2 and 3 $\sigma$ levels from the COSMOS IRAC channel 1
image from Annunziatella et al. (2023), with an applied Gaussian blur
($\sigma=0.8$). We label each source with our catalog ID and its MIRI 5.6
$\mu$m flux.

##  5 Summary & Conclusions

In this paper, we present a study of the number counts and source properties
of sources detected in eight MIRI F560W images with a combined area of 18.4
sq.arcmin and a 5 $\sigma$ depth of $\approx$0.1 $\mu$Jy. Two of our eight
fields overlap with the COSMOS2020 catalog which also allows for a more
detailed look at the source population properties. Below we summarize our key
findings:

  * •

The pipeline reduced MIRI F560W images required further background
subtraction, given strong stripping artifacts. We also note some residual
structure seen in the noise, consistent with tree ring artifacts as noted in
$JWST$ documentation. The photometric zero point needed to be adjusted since
the header keyword in these early images was not using the true measured MIRI
PSF.

  * •

Our study includes 8 widely separated fields, allowing us to explore the
effects of cosmic variance. We used this to construct field-to-field variation
uncertainty on our combined MIRI 5.6 $\mu$m number counts.

  * •

Our number counts have a more pronounced knee, at $\approx$ 2 $\mu$Jy and are
$\approx 2\times$ higher than those computed from the CEERS images (Yang et
al., 2023). This difference however is only at $\approx$1 $\sigma$ level given
our measured field-to-field variation. The observed counts are consistent with
the Cowley et al. (2018) SAM predictions around the knee. These models however
overpredict the observed counts below the knee of the counts.

  * •

We find 84% of the MIRI sources in the COSMOS fields have counterparts in the
COSMOS2020 catalog. They are predominantly at cosmic noon, and often dusty
star-forming galaxies with stellar masses well below $M^{*}$.

  * •

We find 31 very red sources that are securely detected in MIRI, but are not in
the COSMOS2020 catalog. This population has a source density of
$\approx$7/sq.arcmin. They are consistent with being either intermediate
redshift ($z\sim 1$) dusty lower mass galaxies (those with a strong IRAC
channel 1 detection, 10–20%) or high mass, high-$z$ ($z\gtrsim 4$) dusty
galaxies (those with weak or no IRAC channel 1 detections, 80–90%). These MIRI
5.6 $\mu$m only sources will be explored further in a follow-up paper.

All the JWST data used in this paper can be found in MAST:
http://dx.doi.org/10.17909/jsqw-mq02 (catalog 10.17909/jsqw-mq02).

## Acknowledgments

We are grateful to Meredith Stone for providing us with her MIRI 5.6 $\mu$m
counts ahead of publication. We are also grateful to Matthew Ashby for
providing us with a copy of his old IRAC channel 3 counts. Based on
observations with the NASA/ESA/CSA JWST obtained at the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Incorporated, under NASA contract NAS5-03127. Support
for program number JWST-GO-01762 was provided through a grant from the STScI
under grant number JWST-GO-01762.010-A.

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