# Rings whose invertible elements are
weakly nil-clean

Peter Danchev  Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>,
Omid Hasanzadeh  Department of Mathematics, Tarbiat Modares University,
14115-111 Tehran Jalal AleAhmad Nasr, Iran<EMAIL_ADDRESS>,  Arash
Javan  Department of Mathematics, Tarbiat Modares University, 14115-111 Tehran
Jalal AleAhmad Nasr, Iran<EMAIL_ADDRESS>and  Ahmad Moussavi
Department of Mathematics, Tarbiat Modares University, 14115-111 Tehran Jalal
AleAhmad Nasr, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study those rings in which all invertible elements are weakly nil-clean
calling them UWNC rings. This somewhat extends results due to Karimi-Mansoub
et al. in Contemp. Math. (2018), where rings in which all invertible elements
are nil-clean were considered abbreviating them as UNC rings. Specifically,
our main achievements are that the triangular matrix ring ${\rm T}_{n}(R)$
over a ring $R$ is UWNC precisely when $R$ is UNC. Besides, the notions UWNC
and UNC do coincide when $2\in J(R)$. We also describe UWNC $2$-primal rings
$R$ by proving that $R$ is a ring with $J(R)={\rm Nil}(R)$ such that $U(R)=\pm
1+{\rm Nil}(R)$. In particular, the polynomial ring $R[x]$ over some arbitrary
variable $x$ is UWNC exactly when $R$ is UWNC. Some other relevant assertions
are proved in the present direction as well.

###### Key words and phrases:

nil-clean element (ring), weakly nil-clean element (ring), UU-rings, weakly
UU-rings

###### 2010 Mathematics Subject Classification:

16S34, 16U60

##  1\. Introduction and Major Concepts

Everywhere in the current paper, let $R$ be an associative but not necessarily
commutative ring having identity element, usually stated as $1$. Standardly,
for such a ring $R$, the letters $U(R)$, $\rm{Nil}(R)$ and ${\rm Id}(R)$ are
designed for the set of invertible elements (also termed as the unit group of
$R$), the set of nilpotent elements and the set of idempotent elements in $R$,
respectively. Likewise, $J(R)$ denotes the Jacobson radical of $R$, and ${\rm
Z}(R)$ denotes the center of $R$. The ring of $n\times n$ matrices over $R$
and the ring of $n\times n$ upper triangular matrices over $R$ are denoted by
${\rm M}_{n}(R)$ and ${\rm T}_{n}(R)$, respectively. Standardly, a ring is
said to be abelian if each of its idempotents is central, that is, ${\rm
Id}(R)\subseteq{\rm Z}(R)$.

For all other unexplained explicitly notions and notations, we refer to the
classical source [17] or to the cited in the bibliography research sources.
However, for completeness of the exposition and for the reader’s convenience,
we recall the following basic notions.

######  Definition 1.1 ([13]).

Let $R$ be a ring. An element $r\in R$ is said to be nil-clean if there is an
idempotent $e\in R$ and a nilpotent $b\in R$ such that $r=e+b$. Such an
element $r$ is further called strongly nil-clean if the existing idempotent
and nilpotent can be chosen such that $be=eb$. A ring is called nil-clean
(respectively, strongly nil-clean) if each of its elements is nil-clean
(respectively, strongly nil-clean).

######  Definition 1.2 ([12],[3],[6]).

A ring $R$ is said to be weakly nil-clean provided that, for any $a\in R$,
there exists an idempotent $e\in R$ such that $a-e$ or $a+e$ is nilpotent. A
ring $R$ is said to be strongly weakly nil-clean provided that, for any $a\in
R$, $a$ or $-a$ is strongly nil-clean.

######  Definition 1.3 ([4],[11]).

A ring is called UU if all of its units are unipotent,that is, $U(R)\subseteq
1+{\rm Nil}(R)$ (and so $1+{\rm Nil}(R)=U(R)$).

######  Definition 1.4 ([9]).

A ring $R$ is called weakly UU and abbreviated as $WUU$ if $U(R)={\rm
Nil}(R)\pm 1$. This is equivalent to the condition that every unit can be
presented as either $n+1$ or $n-1$, where $n\in{\rm Nil}(R)$.

######  Definition 1.5 ([14]).

A ring $R$ is called UNC if every of its units is nil-clean.

Our key working instrument is the following one.

######  Definition 1.6.

A ring $R$ is called UWNC if every of its units is weakly nil-clean.

In [14], the authors investigated UNC rings, i.e., those rings whose units are
nil-clean. Our plan is to expand this substantially by developing an useful
for this purpose machinery. In fact, we use some non-standard techniques from
ring and matrix theories as well as also compare and distinguish the
established results with these from [9].

It is worth noticing that some closely related background material can also be
found in the sources [10] and [18].

The next constructions that show some proper ring classes inclusions are
worthwhile.

######  Example 1.7.

  1. (i)

Any nil-clean ring is weakly nil-clean, but the converse is not true in
general. For instance, $\mathbb{Z}_{3}$ is weakly nil-clean but is not nil-
clean.

  2. (ii)

Any UU ring is WUU, but the converse is not true in general. For instance,
$\mathbb{Z}$ is a WUU ring but is not UU.

  3. (iii)

Any UU ring and nil-clean ring are UNC, but the converse is not true. For
instance, the direct sum $\mathbb{Z}_{2}[t]\oplus{\rm M}_{2}(\mathbb{Z}_{2})$
is a UNC ring which is neither UU nor nil-clean.

  4. (iv)

Any WUU ring and weakly nil-clean ring are UWNC, but the converse is not true
in general. For instance, the direct sum $\mathbb{Z}\oplus
M_{2}(\mathbb{Z}_{2})$ is a UWNC ring which is neither WUU nor weakly nil-
clean.

  5. (v)

Any UNC ring is UWNC, but the converse is not true in general. For instance,
all of the rings $\mathbb{Z}_{3}$, $\mathbb{Z}_{6}$, $\mathbb{Z}_{3}\oplus{\rm
M}_{2}(\mathbb{Z}_{2})$ are UWNC but are not UNC.

nil-clean weakly nil-clean UNC UWNC UU UNC WUU UWNC

Our further programm is the following: In the next second section, we obtain
some main properties of the newly defined class of UWNC rings – the main
results here are Proposition 2.14 as well as Theorems 2.21, 2.23 and 2.33,
respectively. In the subsequent third section, we explore UWNC group rings –
the main results here are Propositions 3.3 and 3.5, respectively.

##  2\. Basic Properties of UWNC Rings

We start here with the following obvious technicality.

######  Proposition 2.1.

A unit $u$ of a ring $R$ is strongly weakly nil-clean if, and only if,
$u\in\pm 1+{\rm Nil}(R)$. In particular, $R$ is a WUU ring if, and only if,
every unit of $R$ is strongly weakly nil-clean.

We continue our work with the next two technical claims as follows.

######  Proposition 2.2.

Let $R$ be a UWNC ring and $S$ a UNC ring. Then, $R\times S$ is a UWNC ring.

###### Proof.

Choose $(u,v)\in U(R\times S)=U(R)\times U(S)$. Thus, there exist an
idempotent $e\in R$ and a nilpotent $n\in R$ such that $u=e+n$ or $u=-e+n$. We
now differ two cases:

Case I. Write $u=e+n$. Then, we have an idempotent $f\in S$ and a nilpotent
$n^{\prime}\in S$ such that $v=f+n^{\prime}$. Thus,
$(u,v)=(e,f)+(n,n^{\prime})$, where $(e,f)\in{\rm Id}(R\times S)$ and
$(n,n^{\prime})\in{\rm Nil}(R\times S)$.

Case II. Write $u=-e+n$. Then, we have an idempotent $f\in S$ and a nilpotent
$n^{\prime}\in S$ such that $-v=f+n^{\prime}$. Thus,
$(u,v)=-(e,f)+(n,-n^{\prime})$, where $(e,f)\in{\rm Id}(R\times S)$ and
$(n,-n^{\prime})\in{\rm Nil}(R\times S)$.

Therefore, $(u,v)$ is either the sum or difference of a nilpotent and an
idempotent in $R\times S$, whence we get the desired result. ∎

######  Proposition 2.3.

Let $\\{R_{i}\\}$ be a family of rings. Then, the direct product $R=\prod
R_{i}$ of rings $R_{i}$ is UWNC if, and only if, each $R_{i}$ is UWNC and at
most one of them is not UNC.

###### Proof.

($\Rightarrow$). Obviously, each $R_{i}$ is UWNC. Suppose now $R_{i_{1}}$ and
$R_{i_{2}}$ $(i_{1}\neq i_{2})$ are not UNC. Then, there exist some
$u_{i_{j}}\in U(R_{i_{j}})\quad(j=1,2)$ such that $u_{i_{1}}\in U(R_{i_{1}})$
and $-u_{i_{2}}\in U(R_{i_{2}})$ are both not nil-clean. Choosing $u=(u_{i})$,
where $u_{i}=0$ whenever $i\neq i_{j}\quad(j=1,2)$, we infer that $u$ and $-u$
are it not the sum of an idempotent and a nilpotent, as required to get a
contradiction. Consequently, each $R_{i}$ is a UWNC ring as at most one of
them is not UNC.
($\Leftarrow$). Assume that $R_{i_{0}}$ is a UWNC ring and all the others
$R_{i}$ are UNC. So, a simple check gives that $\prod_{i\neq i_{0}}R_{i}$ is
UNC. According to Proposition 2.2, we conclude that $R$ is a UWNC ring. ∎

However, the property of being UWNC is not closed under taking (internal,
external) direct sum as the next construction illustrate.

######  Example 2.4.

A ring $\mathbb{Z}_{3}$ is a UWNC ring, but
$\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ is not UWNC.

Three further helpful affirmations are the following.

######  Corollary 2.5.

Let $L=\prod_{i\in I}R_{i}$ be the direct product of rings $R_{i}\cong R$ and
$|I|\geq 2$. Then, $L$ is a UWNC ring if, and only if, $L$ is a UNC ring if,
and only if, $R$ is a UNC ring.

######  Corollary 2.6.

For any $n\geq 2$, the ring $R^{n}$ is UWNC if, and only if, $R^{n}$ is UNC
if, and only if, $R$ is UNC.

######  Proposition 2.7.

Let $R$ be a UWNC ring. If $T$ is a factor-ring of $R$ such that all units of
$T$ lift to units of $R$, then $T$ is a UWNC ring.

###### Proof.

Suppose that $f:R\rightarrow T$ is a surjective ring homomorphism. Let $v\in
U(T)$. Then, there exists $u\in U(R)$ such that $v=f(u)$ and $u=\pm e+n$,
where $e\in{\rm Id}(R)$ and $n\in{\rm Nil}(R)$. Therefore, we have $v=\pm
f(e)+f(n)$, where $f(e)\in{\rm Id}(T)$ and $f(n)\in{Nil}(T)$, as needed. ∎

We now offer the validity of the following statement.

######  Theorem 2.8.

Let $R$ be a ring and $I$ a nil-ideal of $R$.

  1. (i)

$R$ is a UWNC ring if, and only if, $J(R)$ is nil and $\dfrac{R}{J(R)}$ is a
UWNC ring.

  2. (ii)

$R$ is a UWNC ring if, and only if, $\dfrac{R}{I}$ is a UWNC ring.

###### Proof.

  1. (i)

Let $R$ be a UWNC ring and suppose $x\in J(R)$ and $x\notin{\rm Nil}(R)$.
Since $1+x\in U(R)$, it must be that $1+x=-e+n$, where $n\in{\rm Nil}(R)$ and
$e\in{\rm Id}(R)$, because if $1+x=e+n$, we will have $x\in{\rm Nil}(R)$ that
is a contradiction.

So, $2+x\in{\rm Nil}(R)$. Similarly, since $1+x^{2}\in U(R)$, we deduce that
$2+x^{2}\in{\rm Nil}(R)$. Hence,

$(2+x^{2})-(2+x)=x^{2}-x=-x(1-x)\in{\rm Nil}(R).$

But $1-x\in U(R)$ whence $x\in{\rm Nil}(R)$, a contradiction. Thus, $J(R)$ is
nil.

Now, letting $\bar{u}\in U\left(\bar{R}=\dfrac{R}{J(R)}\right)$, we obtain
$u\in U(R)$, because units lift module $J(R)$. Therefore, $u=\pm e+n$, where
$e\in{\rm Id}(R)$ and $n\in{\rm Nil}(R)$. So, we have
$\bar{u}=\pm\bar{e}+\bar{n}$, where $\bar{e}\in{\rm Id}(\bar{R})$ and
$\bar{n}\in{\rm Nil}(\bar{R})$. Thus, $\dfrac{R}{J(R)}$ is a UWNC ring, as
promised.

Conversely, let $u\in U(R)$. Then, $\bar{u}\in U(\bar{R})$ and write
$\bar{u}=\pm\bar{e}+\bar{n}$, where $\bar{e}\in{\rm Id}(\bar{R})$ and
$\bar{n}\in{\rm Nil}(\bar{R})$. As $J(R)$ is nil, idempotents of
$\dfrac{R}{J(R)}$ can be lifted to idempotents of $R$. So, we can assume that
$e^{2}=e\in R$. Moreover, one inspects that $n\in R$ is nilpotent. Thus, for
some $j\in J(R)$,

$u=\pm e+n+j=\pm e+(n+j)$

is weakly nil-clean, because $n+j\in{\rm Nil}(R)$, as expected.

  2. (ii)

The proof is similar to (i), so we omit the details.

∎

Given a ring $R$ and a bi-module ${}_{R}M_{R}$, the trivial extension of $R$
by $M$ is the ring $T(R,M)=R\oplus M$ with the usual addition and the
following multiplication:
$(r_{1},m_{1})(r_{2},m_{2})=(r_{1}r_{2},r_{1}m_{2}+m_{1}r_{2})$. This is
isomorphic to the ring of all matrices $\begin{pmatrix}r&m\\\
0&r\end{pmatrix}$, where $r\in R$ and $m\in M$ and the usual matrix operation
are used.

As an immediate consequence, we yield:

######  Corollary 2.9.

Let $R$ be a ring and $M$ a bi-module over $R$. Then, the following hold:

  1. (i)

The trivial extension ${\rm T}(R,M)$ is a UWNC ring if, and only if, $R$ is a
UWNC ring.

  2. (ii)

For $n\geq 2$, the quotient-ring $\dfrac{R[x]}{\langle x^{n}\rangle}$ is a
UWNC ring if, and only if, $R$ is a UWNC ring.

  3. (iii)

For $n\geq 2$, the quotient-ring $\dfrac{R[[x]]}{\langle x^{n}\rangle}$ is a
UWNC ring if, and only if, $R$ is a UWNC ring.

###### Proof.

  1. (i)

Set $A={\rm T}(R,M)$ and consider $I:={\rm T}(0,M)$. It is not too hard to
verify that $I$ is a nil-ideal of $A$ such that $\dfrac{A}{I}\cong R$. So, the
result follows directly from Theorem 2.8.

  2. (ii)

Put $A=\dfrac{R[x]}{\langle x^{n}\rangle}$. Considering $I:=\dfrac{\langle
x\rangle}{\langle x^{n}\rangle}$, we obtain that $I$ is a nil-ideal of $A$
such that $\dfrac{A}{I}\cong R$. So, the result follows automatically Theorem
2.8.

  3. (iii)

Knowing that the isomorphism $\dfrac{R[x]}{\langle
x^{n}\rangle}\cong\dfrac{R[[x]]}{\langle x^{n}\rangle}$ is true, point (iii)
follows automatically from (ii).

∎

Consider $R$ to be a ring and $M$ to be a bi-module over $R$. Let

${\rm DT}(R,M):=\\{(a,m,b,n)|a,b\in R,m,n\in M\\}$

with addition defined componentwise and multiplication defined by

$(a_{1},m_{1},b_{1},n_{1})(a_{2},m_{2},b_{2},n_{2})=(a_{1}a_{2},a_{1}m_{2}+m_{1}a_{2},a_{1}b_{2}+b_{1}a_{2},a_{1}n_{2}+m_{1}b_{2}+b_{1}m_{2}+n_{1}a_{2}).$

Then, ${\rm DT}(R,M)$ is a ring which is isomorphic to ${\rm T}({\rm
T}(R,M),{\rm T}(R,M))$. Also, we have

${\rm DT}(R,M)=\left\\{\begin{pmatrix}a&m&b&n\\\ 0&a&0&b\\\ 0&0&a&m\\\
0&0&0&a\end{pmatrix}|a,b\in R,m,n\in M\right\\}.$

We have the following isomorphism as rings: $\dfrac{R[x,y]}{\langle
x^{2},y^{2}\rangle}\rightarrow{\rm DT}(R,R)$ defined by

$a+bx+cy+dxy\mapsto\begin{pmatrix}a&b&c&d\\\ 0&a&0&c\\\ 0&0&a&b\\\
0&0&0&a\end{pmatrix}.$

We, thereby, detect the following.

######  Corollary 2.10.

Let $R$ be a ring and $M$ a bi-module over $R$. Then the following statements
are equivalent:

  1. (i)

$R$ is a UWNC ring.

  2. (ii)

${\rm DT}(R,M)$ is a UWNC ring.

  3. (iii)

${\rm DT}(R,R)$ is a UWNC ring.

  4. (iv)

$\dfrac{R[x,y]}{\langle x^{2},y^{2}\rangle}$ is a UWNC ring.

Another two consequences of interest are the following ones:

######  Corollary 2.11.

Let $R$, $S$ be rings and let $M$ be an $(R,S)$-bi-module. If
$T=\begin{pmatrix}R&M\\\ 0&S\end{pmatrix}$ is a UWNC ring, then both $R,S$ are
UWNC rings. The converse holds provided one of the rings $R$ or $S$ is UNC and
the other is UWNC.

###### Proof.

Given $I:=\left(\begin{array}[]{ll}0&M\\\ 0&0\end{array}\right)$. A routine
inspection shows that this is a nil-ideal in $T$ with $\dfrac{T}{I}\cong
R\times S$. Therefore, the result follows applying Proposition 2.3 and Theorem
2.8. ∎

Let $\alpha$ be an endomorphism of $R$ and $n$ a positive integer. It was
defined by Nasr-Isfahani in [19] the skew triangular matrix ring like this:

${\rm
T}_{n}(R,\alpha)=\left\\{\left.\begin{pmatrix}a_{0}&a_{1}&a_{2}&\cdots&a_{n-1}\\\
0&a_{0}&a_{1}&\cdots&a_{n-2}\\\ 0&0&a_{0}&\cdots&a_{n-3}\\\
\ddots&\ddots&\ddots&\vdots&\ddots\\\
0&0&0&\cdots&a_{0}\end{pmatrix}\right|a_{i}\in R\right\\}$

with addition point-wise and multiplication given by:

$\displaystyle\begin{pmatrix}a_{0}&a_{1}&a_{2}&\cdots&a_{n-1}\\\
0&a_{0}&a_{1}&\cdots&a_{n-2}\\\ 0&0&a_{0}&\cdots&a_{n-3}\\\
\ddots&\ddots&\ddots&\vdots&\ddots\\\
0&0&0&\cdots&a_{0}\end{pmatrix}\begin{pmatrix}b_{0}&b_{1}&b_{2}&\cdots&b_{n-1}\\\
0&b_{0}&b_{1}&\cdots&b_{n-2}\\\ 0&0&b_{0}&\cdots&b_{n-3}\\\
\ddots&\ddots&\ddots&\vdots&\ddots\\\ 0&0&0&\cdots&b_{0}\end{pmatrix}=$
$\displaystyle\begin{pmatrix}c_{0}&c_{1}&c_{2}&\cdots&c_{n-1}\\\
0&c_{0}&c_{1}&\cdots&c_{n-2}\\\ 0&0&c_{0}&\cdots&c_{n-3}\\\
\ddots&\ddots&\ddots&\vdots&\ddots\\\ 0&0&0&\cdots&c_{0}\end{pmatrix},$

where

$c_{i}=a_{0}\alpha^{0}(b_{i})+a_{1}\alpha^{1}(b_{i-1})+\cdots+a_{i}\alpha^{i}(b_{i}),~{}~{}1\leq
i\leq n-1.$

We denote the elements of ${\rm T}_{n}(R,\alpha)$ by
$(a_{0},a_{1},\ldots,a_{n-1})$. If $\alpha$ is the identity endomorphism, then
${\rm T}_{n}(R,\alpha)$ is a subring of upper triangular matrix ring ${\rm
T}_{n}(R)$.

All of the mentioned above guarantee the truthfulness of the following
statement.

######  Corollary 2.12.

Let $R$ be a ring. Then, the following are equivalent:

  1. (i)

$R$ is a UWNC ring.

  2. (ii)

${\rm T}_{n}(R,\alpha)$ is a UWNC ring.

###### Proof.

Choose

$I:=\left\\{\left.\begin{pmatrix}0&a_{12}&\ldots&a_{1n}\\\
0&0&\ldots&a_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&0\end{pmatrix}\right|a_{ij}\in R\quad(i\leq j)\right\\}.$

Then, one easily verifies that $I^{n}=0$ and $\dfrac{{\rm
T}_{n}(R,\alpha)}{I}\cong R$. Consequently, Theorem 2.8 applies to get the
wanted result. ∎

Let $\alpha$ be an endomorphism of $R$. We denote by $R[x,\alpha]$ the skew
polynomial ring whose elements are the polynomials over $R$, the addition is
defined as usual, and the multiplication is defined by the equality
$xr=\alpha(r)x$ for any $r\in R$. Thus, there is a ring isomorphism

$\varphi:\dfrac{R[x,\alpha]}{\langle x^{n}\rangle}\rightarrow{\rm
T}_{n}(R,\alpha),$

given by

$\varphi(a_{0}+a_{1}x+\ldots+a_{n-1}x^{n-1}+\langle
x^{n}\rangle)=(a_{0},a_{1},\ldots,a_{n-1})$

with $a_{i}\in R$, $0\leq i\leq n-1$. So, one finds that ${\rm
T}_{n}(R,\alpha)\cong\dfrac{R[x,\alpha]}{\langle x^{n}\rangle}$, where
$\langle x^{n}\rangle$ is the ideal generated by $x^{n}$.

We, thus, extract the following claim.

######  Corollary 2.13.

Let $R$ be a ring with an endomorphism $\alpha$ such that $\alpha(1)=1$. Then,
the following are equivalent:

  1. (i)

$R$ is a UWNC ring.

  2. (ii)

$\dfrac{R[x,\alpha]}{\langle x^{n}\rangle}$ is a UWNC ring.

  3. (iii)

$\dfrac{R[[x,\alpha]]}{\langle x^{n}\rangle}$ is a UWNC ring.

The next assertion is pivotal.

######  Proposition 2.14.

Let $R$ be a ring. Then, the following are equivalent:

  1. (i)

$R$ is a UNC ring.

  2. (ii)

${\rm T}_{n}(R)$ is a UNC ring for all $n\in\mathbb{N}$.

  3. (iii)

${\rm T}_{n}(R)$ is a UNC ring for some $n\in\mathbb{N}$.

  4. (iv)

${\rm T}_{n}(R)$ is a UWNC ring for some $n\geq 2$.

###### Proof.

(i) $\Rightarrow$ (ii). This follows employing [14, Corollary 2.6].
(ii) $\Rightarrow$ (iii) $\Rightarrow$ (iv). These two implications are
trivial, so we remove the details.
(iv) $\Rightarrow$ (i). Method 1: Let $u\in U(R)$ and choose

$A=\begin{pmatrix}u&&&\ast\\\ &-u_{1}&&\\\ &&\ddots&\\\ 0&&&1\end{pmatrix}\in
U({\rm T}_{n}(R)).$

By hypothesis, we can find an idempotent $\begin{pmatrix}e_{1}&&&\ast\\\
&e_{2}&&\\\ &&\ddots&\\\ 0&&&e_{n}\end{pmatrix}$ and a nilpotent
$\begin{pmatrix}w_{1}&&&\ast\\\ &w_{2}&&\\\ &&\ddots&\\\
0&&&w_{n}\end{pmatrix}$ such that

$A=\pm\begin{pmatrix}e_{1}&&&\ast\\\ &e_{2}&&\\\ &&\ddots&\\\
0&&&e_{n}\end{pmatrix}+\begin{pmatrix}w_{1}&&&\ast\\\ &w_{2}&&\\\ &&\ddots&\\\
0&&&w_{n}\end{pmatrix}.$

It now follows that $u=e_{1}+w_{1}$ or $u=e_{2}-w_{2}$. Clearly, $e_{1}$,
$e_{2}$ are idempotents and $w_{1}$, $w_{2}$ are nilpotents in $R$, thus
proving point (i).

Method 2: Setting $I:=\\{(a_{ij})\in T_{n}(R)|a_{ii}=0\\}$, we obtain that it
is a nil-ideal in ${\rm T}_{n}(R)$ with $\dfrac{{\rm T}_{n}(R)}{I}\cong
R^{n}$. Therefore, Theorem 2.8 and Corollary 2.6 are applicable to get the
pursued result. ∎

We know that the direct sum $\mathbb{Z}_{2}[x]\oplus{\rm
M}_{2}(\mathbb{Z}_{2})$ is a UWNC ring that is neither WUU nor weakly nil-
clean. In this vein, the following example concretely demonstrates an
indecomposable UWNC ring that is neither WUU nor weakly nil-clean.

######  Example 2.15.

Let $R={\rm M}_{n}(\mathbb{Z}_{2})$ with $n\geq 2$, $S=\mathbb{Z}_{2}[x]$ and
$M=S^{n}$. Then, the formal triangular matrix ring $T:=\begin{pmatrix}R&M\\\
0&S\end{pmatrix}$ is an indecomposable UNC ring invoking [14, Example 2.7].
So, $T$ is an indecomposable UWNC ring. But since $R$ is not a WUU ring and
$S$ is not a weakly nil-clean ring, one plainly follows that $T$ is neither a
WUU ring nor a weakly nil-clean ring, as claimed.

It was proved in [9, Proposition 2.25] that any unital subring of a WUU ring
is again a WUU ring. But, curiously, a subring of a UWNC ring may not be a
UWNC ring as the next example shows.

######  Example 2.16.

Let $T={\rm M}_{2}(\mathbb{Z}_{2})$ and $u=\begin{pmatrix}0&1\\\
1&1\end{pmatrix}\in T$. Then, one sees that $u^{3}=1$. Now, let $R$ be the
unital subring of $T$ generated by $u$. Therefore, one calculates that

$R=\\{a1+bu+cu^{2}|a,b,c\in\mathbb{Z}\\}=\\{0,1,u,u^{2},1+u,1+u^{2},1+u+u^{2},u+u^{2}\\}.$

But, we have that $1+u=u^{2}$, $1+u^{2}=u$, $1+u+u^{2}=0$ and $u+u^{2}=1$, so
we deduce $R=\\{0,1,u,u^{2}\\}$. It is now easy to see that ${\rm
Nil}(R)=\\{0\\}$, so that $R$ is reduced. As $u^{2}\neq u$, $u$ is manifestly
not weakly nil-clean utilizing [3, Theorem 20], and so $T$ is a UWNC ring, as
asserted.

######  Proposition 2.17.

The following two statements are valid:

  1. (i)

If $R$ is a weakly nil-clean ring, then ${\rm Z}(R)$ is strongly weakly nil-
clean.

  2. (ii)

If $R$ is a UWNC ring, then ${\rm Z}(R)$ is a WUU ring.

###### Proof.

  1. (i)

Let $a\in{\rm Z}(R)$. Then, $a\in R$ is weakly nil-clean and central, so $a$
is strongly weakly nil-clean in $R$. Thus, $a\pm a^{2}\in{\rm Nil}(R)$ by [5,
Theorem 2.1]. But, $a\pm a^{2}\in{\rm Z}(R)$ so that

$a\pm a^{2}\in{\rm Nil}(R)\cap{\rm Z}(R)\subseteq{\rm Nil}({\rm Z}(R)).$

Hence, ${\rm Z}(R)$ is strongly weakly nil-clean, as required.

  2. (ii)

The proof is analogous to (i).

∎

######  Proposition 2.18.

For any ring $R$, the power series ring $R[[x]]$ is not UWNC.

###### Proof.

Note the principal fact that the Jacobson radical of $R[[x]]$ is not nil (see,
e.g., [17]). Thus, in view of Theorem 2.8, $R[[x]]$ is really not a UWNC ring,
as expected. ∎

######  Lemma 2.19.

Let $R$ be a ring. Then, the following two points are equivalent:

  1. (i)

$R$ is a UNC ring.

  2. (ii)

$R$ is a UWNC ring and $2\in J(R)$.

###### Proof.

(i) $\Longrightarrow$ (ii). Evidently, $R$ is a UWNC ring. Also, we have $2\in
J(R)$ in virtue of [14, Lemma 2.4].
(ii) $\Rightarrow$ (i). Notice that $\dfrac{R}{J(R)}$ is of characteristic
$2$, because $2\in J(R)$, and so $a=-a$ for every $a\in\dfrac{R}{J(R)}$. That
is why, $\dfrac{R}{J(R)}$ is a UNC ring, and thus we can apply [14, Theorem
2.5] since $J(R)$ is nil in view of Theorem 2.8. ∎

Recall that an element $r$ in a ring $R$ is said to be an unipotent if $r-1$
is a nilpotent. The following technical claim is elementary, but rather
applicable in the sequel.

######  Lemma 2.20.

Let $R$ be a ring and let $r\in R$ be the sum of an idempotent and a
nilpotent. If $r^{2}=1$, then $r$ is unipotent.

###### Proof.

Write $r=e+n$ with $e\in{\rm Id}(R)$ and $n\in{\rm Nil}(R)$. Set $f:=1-e$ and
$x:=n(n+1)\in{\rm Nil}(R)$. Taking into account the equality $fn=f(r-e)=fr$,
we compute that

$fx=fn(n+1)=fr(r-e+1)=fr(r+f)=fr^{2}+frf=f+frf,$

and, similarly, that $xf=f+frf$. Hence, $fx=xf$, so that $x$ is a nilpotent
which commutes with $f$, $e$, $n$ and $r$, respectively. Accordingly,

$f=fr^{2}=fr\cdot r=fnr=fx(1+n)^{-1}r=f(1+n)^{-1}r\cdot x$

is a nilpotent and hence $f=0$, as desired. ∎

Our next main result, which sounds quite surprisingly, is the following.

######  Theorem 2.21.

Let $R$ be a ring and $2\in U(R)$. Then, the following two items are
equivalent:

  1. (i)

$R$ is a UWNC ring.

  2. (ii)

$R$ is a WUU ring.

###### Proof.

(ii) $\Rightarrow$ (i). This is pretty obvious, so we leave the argumentation.
(i) $\Rightarrow$ (ii). First, we show that $R$ is an abelian ring. To this
goal, let $e^{2}=e\in R$, and let $a=1-2e$. Then, it is obviously true that
$a^{2}=1$. Since $R$ is UWNC, either $a$ or $-a$ is nil-clean. By virtue of
Lemma 2.20, one has that $a\in 1+{\rm Nil}(R)$ or $a\in-1+{\rm Nil}(R)$. If
$a\in 1+{\rm Nil}(R)$, then $2e\in{\rm Nil}(R)$, and so $e\in{\rm Nil}(R)$.
This implies that $e=0$. If, however, $a\in-1+{\rm Nil}(R)$, then
$2(1-e)\in{\rm Nil}(R)$, whence $1-e\in{\rm Nil}(R)$. This forces that $e=1$.
Therefore, $R$ has only trivial idempotents. Thus, $R$ is abelian, as
asserted.

Now, let $u\in U(R)$, so $u=\pm e+n$, where $e\in{\rm Id}(R)$ and $n\in{\rm
Nil}(R)$. If $u=e+n$, so $e=u-n\in U(R)$, then $e=1$. If, however, $u=-e+n$,
so $e=u-n\in U(R)$, then $e=1$. Therefore, $u\in\pm 1+{\rm Nil}(R)$. Finally,
one concludes that $R$ is WUU, as formulated. ∎

As an immediate consequence, we derive:

######  Corollary 2.22.

Let $R$ be an abelian ring. Then, the following are equivalent:

  1. (i)

$R$ is a UWNC ring.

  2. (ii)

$R$ is a WUU ring.

Appealing to [9], a commutative ring $R$ is a WUU ring if, and only if, so is
$R[x]$. In what follows, we present a generalization of this result.
Standardly, the prime radical ${\rm N}(R)$ of a ring $R$ is defined to be the
intersection of the prime ideals of $R$. It is know that ${\rm N}(R)={\rm
Nil}_{\ast}(R)$, the lower nil-radical of $R$. A ring $R$ is called a
$2$-primal ring if ${\rm N}(R)$ coincides with ${\rm Nil}(R)$. For an
endomorphism $\alpha$ of a ring $R$, $R$ is called $\alpha$-compatible if, for
any $a,b\in R$, $ab=0\Longleftrightarrow a\alpha(b)=0$, and in this case
$\alpha$ is clearly injective.

We now arrive at our third chief result.

######  Theorem 2.23.

Let $R$ be a 2-primal ring and $\alpha$ an endomorphism of $R$ such that $R$
is $\alpha$-compatible. The following issues are equivalent:

  1. (i)

$R[x,\alpha]$ is a UWNC ring.

  2. (ii)

$R[x,\alpha]$ is a WUU ring.

  3. (iii)

$R$ is a WUU ring.

  4. (iv)

$R$ is a UWNC ring.

  5. (v)

$J(R)={\rm Nil}(R)$ and $U(R)=1\pm J(R)$.

###### Proof.

(ii) $\Rightarrow$ (i) and (iii) $\Rightarrow$ (iv). Straightforward.
(i) $\Rightarrow$ (iv). As $\dfrac{R[x,\alpha]}{\langle x\rangle}\cong R$ and
all units of $\dfrac{R[x,\alpha]}{\langle x\rangle}$ are lifted to units of
$R[x,\alpha]$, the implication easily holds.
(ii) $\Longrightarrow$ (iii). We argue as in the proof of (i)
$\Longrightarrow$ (iv).
(iv) $\Rightarrow$ (v). As $R$ is 2-primal, we have ${\rm Nil}(R)\subseteq
J(R)$, so that $J(R)={\rm Nil}(R)$ bearing in mind Theorem 2.8. Let $a\in
U(R)$, so by hypothesis we have $u=\pm e+n$, where $e\in{\rm Id}(R)$ and
$n\in{\rm Nil}(R)=J(R)$. If $u=e+n$, so $e=u-n\in U(R)$, and thus $e=1$. If,
however, $u=-e+n$, so $e=n-u\in U(R)$, and thus $e=1$. Therefore, we receive
$u\in\pm 1+J(R)$, and hence $U(R)=\pm 1+J(R)$, as required.
(v) $\Rightarrow$ (ii). As $R$ is a $2$-primal ring, with the aid of (v) we
have $J(R)={\rm Nil}(R)=\mathrm{Nil}_{\ast}(R)$. Thus, the quotient-ring
$\dfrac{R}{J(R)}$ is a reduced ring. Moreover, it is easy to see that
$\alpha({\rm Nil}(R))\subseteq{\rm Nil}(R)$, so $\alpha(J(R))\subseteq J(R)$
and $\bar{\alpha}:\dfrac{R}{J(R)}\rightarrow\dfrac{R}{J(R)}$, defined by
$\bar{\alpha}(\bar{a})=\overline{\alpha(a)}$, is an endomorphism of
$\dfrac{R}{J(R)}$.

We next show that $\dfrac{R}{J(R)}$ is $\bar{\alpha}$-compatible. That is, we
must show that, for any $a+J(R),b+J(R)\in\dfrac{R}{J(R)}$, the equivalence

$(a+J(R))(b+J(R))=J(R)\Leftrightarrow(a+J(R))\bar{\alpha}(b+J(R))=J(R)$

holds. Equivalently, we have to show that, for any $a,b\in R$, the equivalence
$ab\in{\rm Nil}(R)\Leftrightarrow a\alpha(b)\in{\rm Nil}(R)$ is true. But this
equivalence has been established in the proof of Claims 1 and 2 in [1, Theorem
3.6]. As $\dfrac{R}{J(R)}$ is a reduced factor-ring and also is
$\bar{\alpha}$-compatible, with [7, Corollary 2.12] at hand we have

$U\left(\dfrac{R}{J(R)}[x,\bar{\alpha}]\right)=U\left(\dfrac{R}{J(R)}\right),$

which is equal to $\\{\pm\bar{1}\\}$ by assumption. So,

$\dfrac{R}{J(R)}[x,\bar{\alpha}]\cong\dfrac{R[x,\alpha]}{J(R)[x,\alpha]}$

is a WUU ring. Also, [7, Lemma 2.2] tells us that

$\mathrm{Nil}_{*}(R[x,\alpha])=\mathrm{Nil}_{*}(R)[x,\alpha].$

Therefore,

$J(R)[x,\alpha]=\mathrm{Nil}_{*}(R[x,\alpha]),$

which is manifestly nil. Hence, [9, Proposition 2.5 and 2.6] ensures that
$R[x,\alpha]$ is a WUU ring, as asked for. ∎

As a direct consequence, we deduce:

######  Corollary 2.24.

Let $R$ be a 2-primal ring. Then, the following are equivalent:

  1. (i)

$R$ is a UWNC ring.

  2. (ii)

$R[x]$ is a UWNC ring.

  3. (iii)

$J(R)={\rm Nil}(R)$ and $U(R)=\pm 1+J(R)$.

The following criterion is worthy of documentation.

######  Proposition 2.25.

Suppose $R$ is a commutative ring. Then, $R[x]$ is a UWNC ring if, and only
if, $R$ is a UWNC ring.

###### Proof.

Method 1: For the necessity, let $R[x]$ be a UWNC ring. We know that
$\dfrac{R[x]}{\langle x\rangle}\cong R$. It, therefore, suffices to show that
$\dfrac{R[x]}{\langle x\rangle}$ is a UWNC ring. To this aim, choosing
$u+\langle x\rangle\in U\left(\dfrac{R[x]}{\langle x\rangle}\right)$, we
derive $u\in U(R[x])$, because all units of $\dfrac{R[x]}{\langle x\rangle}$
are lifted to units of $R[x]$. Thus, we write $u=\pm e+n$, where $e\in{\rm
Id}(R[x])$ and $n\in{\rm Nil}(R[x])$. So,

$u+\langle x\rangle=\pm(e+\langle x\rangle)+(n+\langle x\rangle),$

where $e+\langle x\rangle\in{\rm Id}\left(\dfrac{R[x]}{\langle
x\rangle}\right)$ and $n+\langle x\rangle\in{\rm
Nil}\left(\dfrac{R[x]}{\langle x\rangle}\right)$. Consequently, $u+\langle
x\rangle$ is a weakly nil-clean element and $\dfrac{R[x]}{\langle x\rangle}$
is a UWNC ring, as desired.

For the sufficiency, write

$f=a_{0}+a_{1}x+\ldots+a_{n}x^{n}\in U(R[x]),$

so $a_{0}\in U(R)$ and $a_{1},\ldots,a_{n}\in{\rm Nil}(R)$. By hypothesis, we
have $a_{0}=\pm e+n$, where $e\in{\rm Id}(R)$ and $n\in{\rm Nil}(R)$. This
allows us to infer that

$f=(\pm e+n)+a_{1}x+\cdots+a_{n}x^{n}=\pm e+(n+a_{1}x+\cdots+a_{n}x^{n}),$

where $e\in{\rm Id}(R[x])$ and $n+a_{1}x+\cdots+a_{n}x^{n}\in{\rm Nil}(R[x])$.
Finally, $f$ is a weakly nil-clean element, thus establishing the result.

Method 2: It is well know that every commutative ring is a $2$-primal ring,
and hence the result can be deduced from Corollary 2.24. ∎

Incidentally, we are able to prove the following curious statement.

######  Proposition 2.26.

Let $R$ be a ring, and $m,n\geq 1$. If the matrix rings ${\rm M}_{n}(R)$ and
${\rm M}_{m}(R)$ are both UWNC, then so is the triangular matrix ring ${\rm
T}_{n+m}(R)$.

###### Proof.

Let $V\in U({\rm T}_{n+m}(R))$ be the $(n+m)\times(n+m)$ triangular matrix
which we will write in the block decomposition form as follows
$V=\begin{pmatrix}V_{11}&A_{12}\\\ 0&V_{22}\end{pmatrix}$, where $V_{11}\in
U({\rm M}_{n}(R))$, $V_{22}\in U({\rm M}_{m}(R))$ and $A_{12}$ is
appropriately sized rectangular matrices. By hypothesis, there exist
idempotent matrices $E_{1}$, $E_{2}$ and nilpotent matrices $N_{1}$, $N_{2}$
in ${\rm M}_{n}(R)$ and ${\rm M}_{m}(R)$ such that $V_{11}=\pm E_{1}+N_{1}$
and $V_{22}=\pm E_{2}+N_{2}$. Thus, we obtain the decomposition

$\displaystyle\begin{pmatrix}V_{11}&A_{12}\\\ 0&V_{22}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\pm E_{1}+N_{1}&A_{12}\\\ 0&\pm
E_{2}+N_{2}\end{pmatrix}$ $\displaystyle=\pm\begin{pmatrix}E_{1}&0\\\
0&E_{2}\end{pmatrix}+\begin{pmatrix}N_{1}&A_{12}\\\ 0&N_{2}\end{pmatrix}.$

Since $E_{1}$, $E_{2}$ are idempotents, then an easy verification guarantees
that $\begin{pmatrix}E_{1}&0\\\ 0&E_{2}\end{pmatrix}$ is an idempotent. It is
also readily to see that $\begin{pmatrix}N_{1}&A_{12}\\\ 0&N_{2}\end{pmatrix}$
is a nilpotent. Thus, the above decomposition is the desired weakly nil-clean
decomposition. ∎

Let $A$, $B$ be two rings and $M$, $N$ be $(A,B)$-bi-module and $(B,A)$-bi-
module, respectively. Also, we consider the bilinear maps
$\phi:M\otimes_{B}N\rightarrow A$ and $\psi:N\otimes_{A}M\rightarrow B$ that
apply to the following properties.

$Id_{M}\otimes_{B}\psi=\phi\otimes_{A}Id_{M},Id_{N}\otimes_{A}\phi=\psi\otimes_{B}Id_{N}.$

For $m\in M$ and $n\in N$, define $mn:=\phi(m\otimes n)$ and
$nm:=\psi(n\otimes m)$. Now the $4$-tuple $R=\begin{pmatrix}A&M\\\
N&B\end{pmatrix}$ becomes to an associative ring with obvious matrix
operations that is called a Morita context ring. Denote two-side ideals
$Im\phi$ and $Im\psi$ to $MN$ and $NM$, respectively, that are called the
trace ideals of the Morita context (compare with [2] as well).

We now have at our disposal all the ingredients necessary to establish the
following statement.

######  Proposition 2.27.

Let $R=\left(\begin{array}[]{ll}A&M\\\ N&B\end{array}\right)$ be a Morita
context ring such that $MN$ and $NM$ are nilpotent ideals of $A$ and $B$,
respectively. If $R$ is a UWNC ring, then $A$ and $B$ are UWNC rings. The
converse holds provided one of the $A$ or $B$ is UNC and the other is UWNC.

###### Proof.

Canonically, $\dfrac{M}{M_{0}}$ is an
$\left(\dfrac{A}{J(A)},\dfrac{B}{J(B)}\right)$-bi-module and
$\dfrac{N}{N_{0}}$ is a $\left(\dfrac{B}{J(B)},\dfrac{A}{J(A)}\right)$-bi-
module, and this induces a Morita context
$\begin{pmatrix}\dfrac{A}{J(A)}&\dfrac{M}{M_{0}}\\\
\dfrac{N}{N_{0}}&\dfrac{B}{J(B)}\end{pmatrix}$, where the context products are
given by

$(x+M_{0})(y+N_{0})=xy+J(A),(y+N_{0})(x+M_{0})=yx+J(B)$

for all $x\in M$ and $y\in N$. Thus, an appeal to [2, Lemma 4.5], is a
guarantor that

$\dfrac{R}{J(R)}\cong\begin{pmatrix}\dfrac{A}{J(A)}&\dfrac{M}{M_{0}}\\\
\dfrac{N}{N_{0}}&\dfrac{B}{J(B)}\end{pmatrix}.$

However, as $MN$ and $NM$ are nilpotent ideals of $A$ and $B$, respectively,
we then have that $MN\subseteq J(A)$ and $NM\subseteq J(B)$. Therefore, in
view of [20], we argue that $J(R)=\begin{pmatrix}J(A)&M\\\
N&J(B)\end{pmatrix}$ and hence
$\dfrac{R}{J(R)}\cong\dfrac{A}{J(A)}\times\dfrac{B}{J(B)}$. Since $R$ is a
UWNC ring, then the factor $\dfrac{R}{J(R)}$ is a UWNC ring and $J(R)$ is nil
consulting with Theorem 2.8, so it follows that both $\dfrac{A}{J(A)}$ and
$\dfrac{B}{J(B)}$ are UWNC. As $J(R)$ is nil, $J(A)$ and $J(B)$ are nil too.
Thus, $A$ and $B$ are UWNC as well.

Oppositely, assuming that $A$ is a UNC ring and $B$ is a UWNC ring, we
conclude that $\dfrac{R}{J(R)}$ is a UWNC ring by a combination of [14,
Theorem 2.5], Theorem 2.8 and Proposition 2.2. It then suffices to show that
$J(R)$ is nil. To this target, suppose $r=\begin{pmatrix}a&m\\\
n&b\end{pmatrix}\in J(R)$. Then, $a\in J(A)$, $b\in J(B)$. In virtue of
Theorem 2.8, both ideals $J(A)$ and $J(B)$ are nil. Thus, we can find
$n\in\mathbb{N}$ such that $a^{n}=0$ in $A$ and $b^{n}=0$ in $B$. So,

$\begin{pmatrix}a&m\\\ n&b\end{pmatrix}^{n+1}\subseteq\begin{pmatrix}MN&M\\\
N&NM\end{pmatrix}.$

Clearly,

$\begin{pmatrix}MN&M\\\ N&NM\end{pmatrix}^{2}=\begin{pmatrix}MN&(MN)M\\\
(NM)N&NM\end{pmatrix}.$

Moreover, for any $j\in\mathbb{N}$, one easily checks that

$\begin{pmatrix}MN&M\\\ N&NM\end{pmatrix}^{2j}=\begin{pmatrix}MN&(MN)M\\\
(NM)N&NM\end{pmatrix}^{j}=\begin{pmatrix}(MN)^{j}&(MN)^{j}M\\\
(NM)^{j}N&(NM)^{j}\end{pmatrix}.$

By hypothesis, we may assume that $(MN)^{p}=0$ in $A$ and $(NM)^{p}=0$ in $B$.
Therefore,

$\begin{pmatrix}MN&M\\\ N&NM\end{pmatrix}^{2p}=0.$

Consequently, $\begin{pmatrix}a&m\\\ n&b\end{pmatrix}^{2p(n+1)}=0$, and so
$J(R)$ is indeed nil, as desired. ∎

We now state the following.

######  Example 2.28.

Consider the Morita context $R=\begin{pmatrix}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\\
2\mathbb{Z}_{4}&\mathbb{Z}_{4}\end{pmatrix}$, where the context products are
the same as the product in $\mathbb{Z}_{4}$. Then, we claim that $R$ is UWNC.
Since $\mathbb{Z}_{4}$ is obviously UNC, so we are done exploiting Proposition
2.27. This substantiates our claim.

Given a ring $R$ and a central elements $s$ of $R$, the $4$-tuple
$\begin{pmatrix}R&R\\\ R&R\end{pmatrix}$ becomes a ring with addition
component-wise and with multiplication defined by

$\begin{pmatrix}a_{1}&x_{1}\\\
y_{1}&b_{1}\end{pmatrix}\begin{pmatrix}a_{2}&x_{2}\\\
y_{2}&b_{2}\end{pmatrix}=\begin{pmatrix}a_{1}a_{2}+sx_{1}y_{2}&a_{1}x_{2}+x_{1}b_{2}\\\
y_{1}a_{2}+b_{1}y_{2}&sy_{1}x_{2}+b_{1}b_{2}\end{pmatrix}.$

This ring is denoted by ${\rm K}_{s}(R)$. A Morita context
$\begin{pmatrix}A&M\\\ N&B\end{pmatrix}$ with $A=B=M=N=R$ is called a
generalized matrix ring over $R$. It was observed by Krylov in [16] that a
ring $S$ is a generalized matrix ring over $R$ if, and only if, $S={\rm
K}_{s}(R)$ for some $s\in{\rm Z}(R)$. Here $MN=NM=sR$, so $MN\subseteq
J(A)\Longleftrightarrow s\in J(R)$, $NM\subseteq J(B)\Longleftrightarrow s\in
J(R)$, and $MN$, $NM$ are nilpotent $\Longleftrightarrow s$ is a nilpotent.

As three corollaries, we extract:

######  Corollary 2.29.

Let $R$ be a ring and $s\in{\rm Z}(R)\cap{\rm Nil}(R)$. If ${\rm K}_{s}(R)$ is
a UWNC ring, then $R$ is a UWNC ring. The converse holds, provided $R$ is a
UNC ring.

An other construction of interest is the following one.

######  Example 2.30.

As $\mathbb{Z}_{4}$ is a UNC ring, it follows from Corollary 2.29 that the
generalized matrix ring ${\rm
K}_{(2)}(\mathbb{Z}_{4})=\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}|a,b,c,d\in\mathbb{Z}_{4}\\}$ is a UWNC ring.

Following Tang and Zhou (cf. [21]), for $n\geq 2$ and for $s\in{\rm Z}(R)$,
the $n\times n$ formal matrix ring over $R$ defined by $s$, and denoted by
${\rm M}_{n}(R;s)$, is the set of all $n\times n$ matrices over $R$ with usual
addition of matrices and with multiplication defined below:

For $(a_{ij})$ and $(b_{ij})$ in ${\rm M}_{n}(R;s)$,

$(a_{ij})(b_{ij})=(c_{ij}),\quad\text{where}~{}~{}(c_{ij})=\sum
s^{\delta_{ikj}}a_{ik}b_{kj}.$

Here, $\delta_{ijk}=1+\delta_{ik}-\delta_{ij}-\delta_{jk}$, where
$\delta_{jk}$, $\delta_{ij}$, $\delta_{ik}$ are the Kroncker delta symbols.

We, therefore, have the following.

######  Corollary 2.31.

Let $R$ be a ring and $s\in{\rm Z}(R)\cap{\rm Nil}(R)$. If ${\rm M}_{n}(R;s)$
is a UWNC ring, then $R$ is a UWNC ring. The converse holds, provided $R$ is a
UNC ring.

###### Proof.

If $n=1$, then ${\rm M}_{n}(R;s)=R$. So, in this case, there is nothing to
prove. Let $n=2$. By the definition of ${\rm M}_{n}(R;s)$, we have ${\rm
M}_{2}(R;s)\cong{\rm K}_{s^{2}}(R)$. Apparently, $s^{2}\in{\rm Nil}(R)\cap{\rm
Z}(R)$, so the claim holds for $n=2$ with the help of Corollary 2.29.

To proceed by induction, assume now that $n>2$ and that the claim holds for
${\rm M}_{n-1}(R;s)$. Set $A:={\rm M}_{n-1}(R;s)$. Then, ${\rm
M}_{n}(R;s)=\begin{pmatrix}A&M\\\ N&R\end{pmatrix}$ is a Morita context, where

$M=\begin{pmatrix}M_{1n}\\\ \vdots\\\
M_{n-1,n}\end{pmatrix}\quad\text{and}\quad N=(M_{n1}\dots M_{n,n-1})$

with $M_{in}=M_{ni}=R$ for all $i=1,\dots,n-1,$ and

$\displaystyle\psi:N\otimes M\rightarrow N,\quad n\otimes m\mapsto snm$
$\displaystyle\phi:M\otimes N\rightarrow M,\quad m\otimes n\mapsto smn.$

Besides, for $x=\begin{pmatrix}x_{1n}\\\ \vdots\\\ x_{n-1,n}\end{pmatrix}\in
M$ and $y=(y_{n1}\dots y_{n,n-1})\in N$, we write

$xy=\begin{pmatrix}s^{2}x_{1n}y_{n1}&sx_{1n}y_{n2}&\dots&sx_{1n}y_{n,n-1}\\\
sx_{2n}y_{n1}&s^{2}x_{2n}y_{n2}&\dots&sx_{2n}y_{n,n-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
sx_{n-1,n}y_{n1}&sx_{n-1,n}y_{n2}&\dots&s^{2}x_{n-1,n}y_{n,n-1}\end{pmatrix}\in
sA$

and

$yx=s^{2}y_{n1}x_{1n}+s^{2}y_{n2}x_{2n}+\dots+s^{2}y_{n,n-1}x_{n-1,n}\in
s^{2}R.$

Since $s$ is nilpotent, we see that $MN$ and $NM$ are nilpotent too. Thus, we
obtain that

$\frac{{\rm M}_{n}(R;s)}{J({\rm
M}_{n}(R;s))}\cong\frac{A}{J(A)}\times\frac{R}{J(R)}.$

Finally, the induction hypothesis and Proposition 2.27 yield the claim after
all. ∎

A Morita context $\begin{pmatrix}A&M\\\ N&B\end{pmatrix}$ is called trivial,
if the context products are trivial, i.e., $MN=0$ and $NM=0$. We now have

$\begin{pmatrix}A&M\\\ N&B\end{pmatrix}\cong{\rm T}(A\times B,M\oplus N),$

where $\begin{pmatrix}A&M\\\ N&B\end{pmatrix}$ is a trivial Morita context
consulting with [15].

What we can now offer is the following.

######  Corollary 2.32.

If the trivial Morita context $\begin{pmatrix}A&M\\\ N&B\end{pmatrix}$ is a
UWNC ring, then $A$, $B$ are UWNC rings. The converse holds if one of the
rings $A$ or $B$ is UWNC and the other is UNC.

###### Proof.

It is apparent to see that the isomorphisms

$\begin{pmatrix}A&M\\\ N&B\end{pmatrix}\cong{\rm T}(A\times B,M\oplus
N)\cong\begin{pmatrix}A\times B&M\oplus N\\\ 0&A\times B\end{pmatrix}$

are fulfilled. Then, the rest of the proof follows combining Corollary 2.9 and
Proposition 2.3. ∎

We now intend to prove the following.

######  Theorem 2.33.

Let $R$ be a local ring. Then, the following are equivalent:

  1. (i)

$R$ is a UWNC ring.

  2. (ii)

$R$ is a weakly nil-clean ring.

###### Proof.

(i) $\Rightarrow$ (ii). As $R$ is local, one finds that ${\rm
Id}(R)=\\{0,1\\}$, and so $R$ is abelian. Therefore, $R$ is WUU owing to
Corollary 2.22. Also, $R$ is clean, because every local ring is clean. Thus,
$R$ is a clean WUU and we apply [9, Corollary 2.15] to find that $R$ is
strongly weakly nil-clean and so it is weakly nil-clean.
(ii) $\Rightarrow$ (i). It is clear. ∎

We say that $B$ is a unital subring of a ring $A$ if $\emptyset\neq B\subseteq
A$ and, for any $x,y\in B$, the relations $x-y$, $xy\in B$ and $1_{A}\in B$
hold. Let $A$ be a ring and $B$ a unital subring of $A$, and denote by
$R[A,B]$ the set $\\{(a_{1},\ldots,a_{n},b,b,\ldots):a_{i}\in A,b\in B,1\leq
i\leq n\\}$. Then, $R[A,B]$ forms a ring under the usual component-wise
addition and multiplication.

We, thereby, establish the following.

######  Proposition 2.34.

Let $A$ be a ring and a unital subring $B$ of $A$. Then, the following are
equivalent:

  1. (i)

$A$ and $B$ are UWNC.

  2. (ii)

$R[A,B]$ is UWNC.

##  3\. UWNC Group Rings

We begin here with the following simple but useful technicality.

######  Lemma 3.1.

Let $R$ and $S$ be rings and $i:R\rightarrow S$, $\varepsilon:S\rightarrow R$
be ring homomorphisms such that $\varepsilon i=id_{R}$.

  1. (i)

$\varepsilon({\rm Nil}(S))={\rm Nil}(R)$, $\varepsilon(U(S))=U(R)$ and
$\varepsilon({\rm Id}(S))={\rm Id}(R)$.

  2. (ii)

If $S$ is a UWNC ring, then $R$ is a UWNC ring.

  3. (iii)

If $R$ is a UWNC ring and $\text{ker}\varepsilon\subseteq{\rm Nil}(S)$, then
$S$ is a UWNC ring.

###### Proof.

  1. (i)

Clearly, the inclusions $\varepsilon({\rm Nil}(S))\subseteq{\rm Nil}(R)$,
$\varepsilon(U(S))\subseteq U(R)$ and $\varepsilon({\rm Id}(S))\subseteq{\rm
Id}(R)$ are valid. On the other hand, we also have that ${\rm
Nil}(R)=\varepsilon i({\rm Nil}(R))\subseteq\varepsilon({\rm Nil}(S))$,
$U(R)=\varepsilon i(U(R))\subseteq\varepsilon(U(S))$ and ${\rm
Id}(R)=\varepsilon i({\rm Id}(R))\subseteq\varepsilon({\rm Id}(S))$.

  2. (ii)

Method 1: Let $S$ be a UWNC ring. Choose $u\in U(R)=\varepsilon(U(S))$, so
$u=\varepsilon(v)$, where $v\in U(S)$. Thus, we can write $v=\pm e+q$, where
$e\in{\rm Id}(S)$, $q\in{\rm Nil}(S)$. Therefore,

$u=\varepsilon(v)=\varepsilon(\pm e+q)=\pm\varepsilon(e)+\varepsilon(q),$

where $\varepsilon(e)\in{\rm Id}(R)$ and $\varepsilon(q)\in{\rm Nil}(R)$
exploiting (i).

Method 2: Let $S$ be a UWNC ring. Hence, $U(S)=\pm{\rm Id}(S)+{\rm Nil}(S)$,
whence by (i) one has that

$U(R)=\varepsilon(U(S))=\varepsilon(\pm{\rm Id}(S)+{\rm
Nil}(S))=\pm\varepsilon({\rm Id}(S))+\varepsilon({\rm Nil}(S))=\pm{\rm
Id}(R)+{\rm Nil}(R),$

as required.

  3. (iii)

If $R$ is a UWNC ring, point (i) enables us that

$U(S)=\varepsilon^{-1}(U(R))=\varepsilon^{-1}(\pm{\rm Id}(R)+{\rm
Nil}(R))=\pm{\rm Id}(S)+{\rm Nil}(S)+\text{ker}\varepsilon=\pm{\rm Id}(S)+{\rm
Nil}(S),$

as required.

∎

######  Remark 3.2.

It is a routine technical exercise to see that Lemma 3.1 (2) and (3) is true
also for WUU rings.

Suppose now that $G$ is an arbitrary group and $R$ is an arbitrary ring. As
usual, $RG$ stands for the group ring of $G$ over $R$. The homomorphism
$\varepsilon:RG\rightarrow R$, defined by $\varepsilon(\displaystyle\sum_{g\in
G}a_{g}g)=\displaystyle\sum_{g\in G}a_{g}$, is called the augmentation map of
$RG$ and its kernel, denoted by $\Delta(G)$, is called the augmentation ideal
of $RG$.

######  Proposition 3.3.

Let $R$ be a ring and $G$ a group. If $RG$ is a UWNC ring, then $R$ is a UWNC
ring. The converse holds if $\Delta(G)\subseteq{\rm Nil}(RG)$.

###### Proof.

Let us consider the inclusion $i:R\rightarrow RG$, given by
$i(r)=\displaystyle\sum_{g\in G}a_{g}g$, where $a_{1G}=r$ and $a_{g}=0$
provided $g\neq 1_{G}$. It is easy to check that the map $i$ is a ring
monomorphism and thus $R$ can also be viewed as a subring of $RG$.
Furthermore, it is only enough to apply Lemma 3.1 (ii) to get the claim. ∎

######  Corollary 3.4.

Let $R$ be a ring and $G$ a group. If $RG$ is a WUU ring, then $R$ is a WUU
ring. The converse holds if $\Delta(G)\subseteq{\rm Nil}(RG)$.

###### Proof.

It follows at once by Proposition 3.3 and Lemma 3.1. ∎

A group $G$ is called locally finite if every finitely generated subgroup of
$G$ is finite. Let $p$ be a prime number. A group $G$ is called a $p$-group if
the order of each element of $G$ is a power of $p$.

We finish off our results with the following statement.

######  Proposition 3.5.

Let $R$ be a UWNC ring with $p\in{\rm Nil}(R)$ and let $G$ be a locally finite
$p$-group, where $p$ is a prime. Then, $RG$ is a UWNC ring.

###### Proof.

Referring to [8, Proposition 16], one verifies that $\Delta(G)$ is nil. Now,
the assertion follows from the obvious isomorphism $\dfrac{RG}{\Delta(G)}\cong
R$ and Theorem 2.8. ∎

######  Remark 3.6.

It is easily seen that Proposition 3.5 holds also for WUU rings.

##  4\. Open Questions

We close the work with the following challenging conjectures and problems.

A ring $R$ is called uniquely weakly nil-clean, provided that $R$ is a weakly
nil-clean ring in which every nil-clean element is uniquely nil-clean (see
[6]).

######  Conjecture 4.1.

A ring R is a WUU ring if, and only if, every unit of $R$ is uniquely weakly
nil-clean.

######  Conjecture 4.2.

A ring $R$ is a strongly weakly nil-clean if, and only if, it is a semi-potent
WUU ring.

######  Problem 4.3.

Is a clean, UWNC ring a weakly nil-clean ring?

######  Problem 4.4.

Characterize semi-perfect UWNC rings. Are they weakly nil-clean?

######  Problem 4.5.

Suppose that $R$ is a ring and $n\in\mathbb{N}$. Find a criterion when the
full $n\times n$ matrix ring ${\rm M}_{n}(R)$ is UWNC.

Funding: The work of the first-named author, P.V. Danchev, is partially
supported by the project Junta de Andalucía under Grant FQM 264, and by the
BIDEB 2221 of TÜBÍTAK.

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