SIT-ring and SITT-ring properties in amalgamated and

bi-amalgamated algebras along ideals

A. Aruldoss and C. Selvaraj

Department of Mathematics, Periyar University

Salem - 636 011, Tamilnadu, India.

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

AMS Subject classification (2010): 13A15, 13D05.

Abstract: Let $f:A\longrightarrow B$ and $g:A\longrightarrow C$ be two ring
homomorphisms and let $J$ and $J^{\prime}$ be two ideals of $B$ and $C$,
respectively, such that $f^{-1}(J)=g^{-1}(J^{\prime})$. In this paper, we give
a characterization for the amalgamation of $A$ with $B$ along $J$ with respect
to $f$ (denoted by $A\bowtie^{f}J$) to be a SIT-ring and also we give a
characterization for the bi-amalgamation of $A$ with $(B,C)$ along
$(J,J^{\prime})$ with respect to $(f,g)$ (denoted by
$A\bowtie^{f,g}(J,J^{\prime})$) to be a SITT-ring. We also give some
characterizations for strong weakly SIT-rings.

Keywords: idempotent, tripotent, amalgamated duplication, amalgamated algebra
along an ideal, bi-amalgamated algebra along an ideal.

##  1 Introduction

Throughout the paper, $R$ denotes an associative ring with identity. An
element $a$ in a ring is an idempotent if $a^{2}=a$. Idempotents are key
elements in understanding the structure of a ring. When a ring has a rich
supply of idempotents, one would usually expect something nice about the
structure of the ring. An extreme case is the notion of Boolean rings, in
which every element is an idempotent. Boolean rings are impotant in algebra
and have applications in other areas. A ring $R$ is weakly Boolean if for any
$a\in R$, either $a$ or $-a$ is an idempotent.

In 2016, Zhiling Ying, Tamer Kosan and Yiqiang Zhou [17] investigated that
rings for which every element is a sum of an idempotent and a tripotent that
commute and proved that every element of a ring $R$ is a sum of an idempotent
and a tripotent that commute and $2\in J(R)$ if and only if $R$ has the
idetity $x^{6}=x^{4}$ and $2\in J(R)$ if and only if $R/J(R)$ is Boolean and
$j^{2}=2j$ for all $j\in J(R)$ if and only if $R/J(R)$ is Boolean and $U(R)$
is a group of exponent 2. It can be shown that every element of a ring is a
sum of an idempotent and a tripotent that commute iff $R\cong A\times B$,
where $A/J(A)$ is Boolean with $U(A)$ a group of exponent 2, and $B$ is a
subdirect product of $\mathbb{Z}_{3}^{\prime}s$. For any idempotent $e$, both
$e$ and $-e$ are tripotents, i.e., the elements equal to their cubes.

Let $A$ and $B$ be two commutative rings with unity, let $J$ be an ideal of
$B$ and let $f:A\rightarrow B$ be a ring homomorphism. In this setting, we can
consider the following subring of $A\times B$:

$A\bowtie^{f}J:=\\{(a,f(a)+j)\mid a\in A,j\in J\\}$

called _the amalgamation of $A$ with $B$ along $J$ with respect to $f$_
(introduced and studied by D’Anna, Finocchiaro, and Fontana in [6, 7]). This
construction is a generalization of _the amalgamated duplication of a ring
along an ideal_ (introduced and studied by D’Anna and Fontana in [3, 4, 5]).
Moreover, other classical constructions (such as the $A+XB[X]$, $A+XB[[X]]$,
and the $D+M$ constructions) can be studied as particular cases of the
amalgamation ([6, Examples 2.5 and 2.6]) and other classical constructions,
such as the Nagata’s idealization (cf. [16, page 2]), and the CPI extensions
(in the sense of Boisen and Sheldon [8]) are strictly related to it ([6,
Example 2.7 and Remark 2.8]). On the other hand, the amalgamation
$A\bowtie^{f}J$ is related to a construction proposed by Anderson in [2] and
motivated by a classical construction due to Dorroh [10], concerning the
embedding of a ring without identity in a ring with identity. An ample
introduction on the genesis of the notion of amalgamation is given in [6,
Section 2]. Also, the authors consider the iteration of the amalgamation
process, giving some geometrical applications of it.

One of the key tools for studying $A\bowtie^{f}J$ is based on the fact that
the amalgamation can be studied in the frame of pullback constructions [6,
Section 4]. This point of view allows the authors in [6, 7] to provide an
ample description of various properties of $A\bowtie^{f}J$, in connection with
the properties of $A$, $J$ and $f$. Namely, in [6], the authors studied the
basic properties of this construction (e.g., characterizations for
$A\bowtie^{f}J$ to be a Noetherian ring, an integral domain, a reduced ring)
and they characterized those distinguished pullbacks that can be expressed as
an amalgamation. Moreover, in [7], they pursue the investigation on the
structure of the rings of the form $A\bowtie^{f}J$, with particular attention
to the prime spectrum, to the chain properties and to the Krull dimension.

Let $\alpha:A\longrightarrow C$, $\beta:A\longrightarrow C$ and
$f:A\longrightarrow B$ be ring homomorphisms. In the aforementioned papers [6,
7], the authors studied amalgamated algebras within the frame of pullback
$\alpha\times\beta$ such that $\alpha=\beta\circ f$ [6, Proposition 4.2 and
4.4]. In this motivation, the authors created the new constructions, called
bi-amalgamated algebras which arise as pullbacks $\alpha\times\beta$ such that
the following diagram of ring homomorphims

$C$$D$$A$$B$$f$$g$$\alpha$$\beta$

is commutative with $\alpha\circ\pi_{B}(\alpha\times\beta)=\alpha\circ f(A)$,
where $\pi_{B}$ denotes the canonical projection of $B\times C$ over $B$.
Namely, let $f:A\longrightarrow B$ and $g:A\longrightarrow C$ be two ring
homomorphisms and let $J$ and $J^{\prime}$ be two ideals of $B$ and $C$,
respectively, such that $f^{-1}(J)=g^{-1}(J^{\prime})$. The bi-amalgamation of
$A$ with $(B,C)$ along $(J,J^{\prime})$ with respect to $(f,g)$ is the subring
of $B\times C$ given by

$A\bowtie^{f,g}(J,J^{\prime}):=\\{(f(a)+j,g(a)+j^{\prime})\rvert a\in
A,(j,j^{\prime})\in J\times J^{\prime}\\}$

In this paper, the second section we introduce a class of rings which
generalizes the so-called SIT-rings namely, weakly SIT-rings and SITT-rings. A
ring $R$ is called (strongly) weakly SIT-ring if every element of $R$ is a sum
or difference of an idempotent and a tripotent (that commute) and a ring $R$
is called a SITT-ring if every element is a sum of an idempotent and two
tripotents. Clearly, SIT-rings are weakly SIT-rings and SITT-rings, but the
converse is not true in general. We give some characterizations of strongly
weakly SIT-rings. Finally, we give some properties of a SITT-rings are
presented. The third section, we give a characterization for $A\bowtie^{f}J$
to be (uniquely) SIT-ring. The fourth section investigates the
characterization for $A\bowtie^{f,g}(J,J^{\prime})$ to be (uniquely) SITT-
ring.

In what follows, $\mathbb{Z}$ denote the ring of integers and for a positive
integer $n$, $\mathbb{Z}_{n}$ is the ring of integers modulo $n$. We write
$M_{n}(R)$ and $T_{n}(R)$ stands for the $n\times n$ matrix ring and $n\times
n$ upper triangular matrix ring, respectively, over $R$. In addition, we write
$R[x]$, $R[[x]]$, $Id(R)$, $Tr(R)$, $Nil(R)$ and $J(R)$ for the polynomial
ring, the power series ring over a ring $R$, the set of all idempotent
elements, set of all tripotent elements of $R$, set of all nilpotent elements
and the Jacobson radical of $R$, respectively.

##  2 SIT-ring and Weakly SIT-ring

We start with a definition.

######  Definition 2.1.

A ring is said to be a (strong) SIT-ring if every element is a sum of an
idempotent and a tripotent (that commute).

######  Proposition 2.2.

The class of SIT-rings is closed under direct product and homomorphic images.

######  Proposition 2.3.

If $R$ is a SIT-ring then $R[x]$ is never a SIT-ring.

###### Proof.

In view of proof of [1, Proposition 12], $Id(R)=Id(R[x])$. Clearly,
$Tr(R)=Tr(R[x])$. Hence $X$ is not a sum of an idempotent and a tripotent. So
$R[x]$ is not a SIT-ring. ∎

######  Proposition 2.4.

Let $R$ be a commutative ring. If $R[[x]]$ is a SIT-ring then $R$ is a SIT-
ring.

###### Proof.

By Proposition 2.2, $R[[x]]$ a SIT-ring gives that $R=R[[x]]/(x)$ is a SIT-
ring. ∎

The converse of the above Proposition need not be true.

Let $R$ be a ring and $\eta:R\longrightarrow R$ be a surjective ring
homomorphism. Then set of all power series in $x$ with coefficients form $R$
with multiplication defined by $xr=\eta(r)x$ for all $r\in R$. It is called
the ring of skew formal power series over $R$ and it is denoted by
$R[[x,\eta]]$. In particular, $R[[x]]=R[[x,1_{R}]]$ is the ring of formal
power series over $R$.

######  Proposition 2.5.

Let $R$ be a ring. If $R[[x,\eta]]$ is a SIT-ring then $R$ is a SIT-ring.

###### Proof.

By Proposition 2.2, $R[[x,\eta]]$ is a SIT-ring, and this gives that
$R=R[[x,\eta]]/(x)$ is a SIT-ring. ∎

######  Remark 2.6.

Matrix ring over a SIT-ring is not a SIT-ring.

######  Example 2.7.

Let $\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$ be two SIT-rings. But the matrix
rings $M_{2}(\mathbb{Z}_{2})$ and $M_{2}(\mathbb{Z}_{3})$ are not SIT-rings
because the element $\begin{pmatrix}\bar{0}&\bar{1}\\\
\bar{1}&\bar{0}\end{pmatrix}$ in $M_{2}(\mathbb{Z}_{2})$ is not a sum of an
idempotent and a tripotent.

######  Proposition 2.8.

Let $e^{2}=e\in R$. If $eR$ is a SIT-ring then $eRe$ is a SIT-ring.

###### Proof.

Let $\phi:eR\longrightarrow eRe$ is defined by $\phi(x)=xe$. It is easy to see
that $\phi$ is an epimorphism of rings, $ker\phi=eR(1-e)$, and
$eR/ker\phi\cong eRe$. By virtue of Proposition 2.2, $eRe$ is a SIT-ring. ∎

######  Theorem 2.9.

Let $R$ be a ring with $2\in J(R)$ and $U(R)$ forms a group of exponent 2.
Then $R$ is strong SIT-ring if and only if $R$ is strongly nil clean.

###### Proof.

$\Rightarrow$ In view of [17, Theorem 3.6], $R/J(R)$ is Boolean. For $j\in
J(R)$, we have $(1-j)^{2}=1$ by hypothesis, and $j^{2}=2j$. Replacing $j$ by
$j(j+1)$, we have $(j(1+j))^{2}=2j(1+j)$. We infer that $j(1+j)j=2j$; that is,
$j^{2}+j^{3}=2j$. It follows that $j^{3}=0$. Hence, $J(R)$ is nil. In view of
[12, Theorem 2.7], $R$ is strongly nil clean.

$\Leftarrow$ For any $a\in R$, there exist $b\in Nil(R)$ and $e^{2}=e$ such
that $eb=be$ and $a-1=e+b$. By hypothesis, we have $(1+b)^{2}=1$, so $1+b$ is
a tripotent. Hence, $a=e+(1+b)$ is a sum of an idempotent and a tripotent that
commute. So $R$ is a strong SIT-ring. ∎

If $U(R)$ does not form a group of exponent $2$, the converse need not be
true. We consider the following example.

######  Example 2.10.

Let
$\mathbb{Z}_{4}[i]=\mathbb{Z}_{4}[x]/(x^{2}-1)=\\{a+bi|a,b\in\mathbb{Z}_{4},i^{2}=-1\\}$
and the Jacobson radical
$J(\mathbb{Z}_{4}[i])=\\{a+bi|a,b\in\mathbb{Z}_{4},a^{2}-b^{2}\in
J(\mathbb{Z}_{4}),i^{2}=-1\\}$. Thus
$J(\mathbb{Z}_{4}[i])=\\{\bar{0},\bar{2}i,\bar{1}+i,\bar{1}+\bar{3}i,\bar{2},\bar{2}+\bar{2}i,\bar{3}+i,\bar{3}+\bar{3}i\\}$.
It is easy to check that $J(\mathbb{Z}_{4}[i])$ is nil. Hence,
$N(\mathbb{Z}_{4}[i])=J(\mathbb{Z}_{4}[i])$ and
$\mathbb{Z}_{4}[i]/J(\mathbb{Z}_{4}[i])$ is Boolean. In view of [12, Theorem
2.7], $\mathbb{Z}_{4}[i]$ is strongly nil clean. In light of Theorem 2.9,
$\mathbb{Z}_{4}[i]$ is not a SIT-ring because $U(\mathbb{Z}_{4}[i])$ is not a
group of exponent 2.

######  Example 2.11.

Let $R=\mathbb{Z}_{4}[x]/(x^{2})=\\{a+bx|a,b\in\mathbb{Z}_{4}\\}$. Then set of
all idempotents of $R$ are $\\{\bar{0},\bar{1}\\}$ and set of all tripotents
of $R$ are $\\{\bar{0},\bar{1},\bar{3},\bar{1}+\bar{2}x,\bar{3}+\bar{2}x\\}$.
According to the definition 2.1, $R$ is not a SIT-ring because the element
$\bar{1}+x\in R$ is not a sum of an idempotent and a tripotent.

######  Example 2.12.

Let $\mathbb{Z}_{(3)}=\\{\frac{m}{n}|m,n\in\mathbb{Z}_{3},3\nmid n\\}$. Then
$\mathbb{Z}_{(3)}$ is a local ring with the Jacobson radical
$J(\mathbb{Z}_{(3)})=3\mathbb{Z}_{(3)}$. Clearly,
$\frac{2}{1}\in\mathbb{Z}_{(3)}$ satisfies
$(\frac{2}{1})^{2}-\frac{2}{1}=\frac{2}{1}\notin J(\mathbb{Z}_{(3)})$; hence
$\mathbb{Z}_{(3)}/J(\mathbb{Z}_{(3)})$ is not Boolean. In view of [17, Theorem
3.6], $\mathbb{Z}_{(3)}$ is not a SIT-ring.

We now introduce the following definition.

######  Definition 2.13.

A ring is said to be (strong) weakly SIT-ring if every element is a sum or
difference of an idempotent and a tripotent (that commute).

######  Proposition 2.14.

The class of (strong) weakly SIT-rings are closed under homomorphic images and
finite direct products . Any factor ring of a (strong) weakly SIT-ring is a
(strongly) weakly SIT-ring.

######  Lemma 2.15.

If $R$ is a ring for which $5=e+f$ or $5=e-f$ where $e^{2}=e$ and $f^{3}=f$,
then $120=0$.

###### Proof.

From $5=e+f$, we see $ef=fe$, so $25=(e+f)^{2}=e+2ef+f^{2}$. Thus,
$4(e+f)=20=25-5=(e+2ef+f^{2})-(e+f)$ = $2ef+f^{2}-f$. It follows that
$4e+5f-2ef-f^{2}=0$. Thus, $0=(4e+5f-2ef-f^{2})ef^{2}$ = $4ef^{2}+5ef-2ef-
ef^{2}=3ef^{2}+3ef$. So, $120=5^{3}-5$ $=(e+f)^{3}-5$ $=(e+3ef+3ef^{2}+f)-5$
$=[(e+f)-5]+[3ef+3ef^{2}]$ $=0$.

From $5=e-f$, so $25=(e-f)^{2}=e-2ef+f^{2}$. Thus,
$4(e-f)=20=25-5=(e-2ef+f^{2})-(e-f)$ = $-2ef+f^{2}+f$. It follows that
$4e-5f+2ef-f^{2}=0$. Thus, $0=(4e-5f+2ef-f^{2})ef^{2}$ = $3ef^{2}-3ef$. So,
$120=5^{3}-5$ $=(e-f)^{3}-5$ $=(e-3ef+3ef^{2}-f)-5$ $=[(e-f)-5]+[3ef^{2}-3ef]$
$=0$. ∎

We now decompose (strong) weakly SIT-ring into product of (strong) weakly SIT-
rings.

######  Lemma 2.16.

A ring $R$ is a (strong) weakly SIT-ring if and only if $R\cong R_{1}\times
R_{2}\times R_{3}$ where $R_{1},R_{2}$ and $R_{3}$ are (strong) weakly SIT-
rings, $2^{3}=0$ in $R_{1}$, $3=0$ in $R_{2}$ and $5=0$ in $R_{3}$.

###### Proof.

The sufficiency is clear by Proposition 2.14. For the necessity, assume that
$R$ is a (strong) weakly SIT-ring. Then, by Lemma 2.15 , $2^{3}.3.5=0$. Thus,
$2^{3}R\bigcap 3R\bigcap 3R=0$ and $R=2^{3}R+3R+5R$. By the Chinese Remainder
Theorem, $R\cong R/2^{3}R\times R/3R\times R/5R$. Let $R_{1}=R/2^{3}R$,
$R_{2}=R/3R$ and $R_{3}=R/5R$. Then $R_{1}$, $R_{2}$ and $R_{3}$ are (strong)
weakly SIT-rings by Proposition 2.14 with $2^{3}=0$ in $R_{1}$, $3=0$ in
$R_{2}$ and $5=0$ in $R_{3}$, and $R\cong R_{1}\times R_{2}\times R_{3}$. ∎

######  Theorem 2.17.

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

  1. 1.

$R$ is strongly weakly nil-clean;

  2. 2.

For any $a\in R$, $a\pm a^{2}$ is nilpotent;

  3. 3.

For any $a\in R$, there exists an idempotent $e\in\mathbb{Z}[x]$ such that
$a\pm e$ is nilpotent;

###### Proof.

This was proved in [9, Theorem 2.1] . ∎

######  Theorem 2.18.

Let $R$ be a ring. Then $R$ is strongly weakly nil-clean iff $R/J(R)$ is
weakly Boolean and $J(R)$ is nil.

###### Proof.

This was proved in [9, Theorem 3.2]. ∎

Following [9], an element of a ring is called (strongly) weakly nil clean if
it is the sum or difference of a nilpotent and an idempotent (that commute),
and the ring is called (strongly) weakly nil clean if each of its elements is
(strongly) weakly nil clean.

######  Theorem 2.19.

Let $R$ be a ring with $2\in J(R)$. Then the following statements are
equivalent.

  1. 1.

$R$ is a strong weakly SIT-ring;

  2. 2.

$R$ is a strong weakly SIT-ring with $2^{3}=0$;

  3. 3.

$R$ has the identity $x^{6}=x^{4}$;

  4. 4.

$R/J(R)$ is weakly Boolean and $j^{2}=\pm 2j$ for all $j\in J(R)$;

  5. 5.

$R/J(R)$ is weakly Boolean and $U(R)$ is a group of exponent $2$;

###### Proof.

(1) $\Rightarrow$ (2). This is clear by Lemma 2.16.

(2) $\Rightarrow$ (3). For $a\in R$, write $a=e+f$ or $a=e-f$ where $e^{2}=e$,
$f^{3}=f$, and $ef=fe$. Then

$\displaystyle a^{4}$ $\displaystyle=$
$\displaystyle(e+f)^{4}=e^{4}+4e^{3}f+6e^{2}f^{2}+4ef^{3}+f^{4}$
$\displaystyle=$ $\displaystyle e+4ef+6ef^{2}+4ef+f^{2}$ $\displaystyle=$
$\displaystyle e+8ef+6ef^{2}+f^{2}$ $\displaystyle=$ $\displaystyle
e+6ef+f^{2}$ $\displaystyle a^{6}$ $\displaystyle=$ $\displaystyle
a^{4}a^{2}=(e+6ef^{2}+f^{2})(e+2ef+f^{2})$ $\displaystyle=$
$\displaystyle(e+6ef^{2}+ef^{2})+(2ef+12ef+2ef)+(ef^{2}+6ef^{2}+f^{2})$
$\displaystyle=$ $\displaystyle e+6ef^{2}+f^{2}$

$\displaystyle a^{4}$ $\displaystyle=$
$\displaystyle(e-f)^{4}=e^{4}-4e^{3}f+6e^{2}f^{2}-4ef^{3}+f^{4}$
$\displaystyle=$ $\displaystyle e-4ef+6ef^{2}-4ef+f^{2}$ $\displaystyle=$
$\displaystyle e-8ef+6ef^{2}+f^{2}$ $\displaystyle=$ $\displaystyle
e+6ef+f^{2}$ $\displaystyle a^{6}$ $\displaystyle=$ $\displaystyle
a^{4}a^{2}=(e+6ef^{2}+f^{2})(e-2ef+f^{2})$ $\displaystyle=$
$\displaystyle(e+6ef^{2}+ef^{2})+(-2ef-12ef-2ef)+(ef^{2}+6ef^{2}+f^{2})$
$\displaystyle=$ $\displaystyle e+6ef^{2}+f^{2}$

So $a^{6}=a^{4}$. Hence, $R$ has the identity $x^{6}=x^{4}$.

(3) $\Rightarrow$ (4). For $j\in J(R)$ , we have $(1-j)^{6}=(1-j)^{4}$, so
$(1-j)^{2}=1$ as $(1-j)^{4}\in U(R)$. It follows that $j^{2}=2j$. Hence, we
have proved that $j^{2}=2j$ for all $j\in J(R)$ and we have
$(1+j)^{6}=(1+j)^{4}$, so $(1+j)^{2}=1$ as $(1+j)^{4}\in U(R)$. It follows
that $j^{2}=-2j$. Hence, we have proved that $j^{2}=-2j$ for all $j\in J(R)$.
From $2^{6}=2^{4}$, we obtain $2^{4}3=0$. As $2\in J(R)$, $3\in U(R)$, so we
infer $2^{4}=0$. For $a\in R$, we have $a^{6}=a^{4}$, so
$(a-a^{2})^{4}=a^{4}(1-a)^{4}=a^{4}(1-4a+6a^{2}-4a^{3}+a^{4})=a^{4}-4a^{5}+6a^{6}-4a^{7}+a^{8}=a^{4}-4a^{5}+6a^{4}-4a^{5}+a^{4}=8(a^{4}-a^{5})$,
which is nilpotent as $2$ is nilpotent. Thus, $a-a^{2}$ is nilpotent and
$(a+a^{2})^{4}=a^{4}(1+a)^{4}=a^{4}(1+4a+6a^{2}+4a^{3}+a^{4})=a^{4}+4a^{5}+6a^{6}+4a^{7}+a^{8}=a^{4}+4a^{5}+6a^{4}+4a^{5}+a^{4}=8(a^{4}+a^{5})$,
which is nilpotent as $2$ is nilpotent. Thus, $a+a^{2}$ is nilpotent. By
Theorem 2.17, there exists $e^{2}=e$ such that $ea=ae$ and $a\pm e$ is
nilpotent. This shows that $a=e+(a-e)$ or $-a=e-(a+e)$ is strongly weakly nil
clean. Therefore, $R$ is strongly weakly nil clean. In light of Theorem 2.18,
$R/J(R)$ is weakly Boolean.

(4) $\Rightarrow$ (5). For $u\in U(R)$ , we have $u^{2}\pm u\in J(R)$ since
$R/J(R)$ is weakly Boolean. Then $u\in 1\pm J(R)$; hence, $U(R)=1\pm J(R)$.
Write $u=1\pm j$ for $j\in J(R)$. Then $u^{2}=(1-j)^{2}=1-2j+j^{2}=1$, as
$j^{2}=2j$ and $u^{2}=(1+j)^{2}=1+2j+j^{2}=1$, as $j^{2}=-2j$. Hence, $U(R)$
is a group of exponent 2.

(5) $\Rightarrow$ (1). For $j\in J(R)$, we have $(1-j)^{2}=1$, so $j^{2}=2j$.
Replacing $j$ by $j(1+j)$, we have $(j(1+j))^{2}=2j(1+j)$. We infer that
$j(1+j)j=2j$; that is $j^{2}+j^{3}=2j$. It follows that $j^{3}=0$ and we have
$(1+j)^{2}=1$, so $j^{2}=-2j$. Replacing $j$ by $j(1-j)$, we have
$(j(1-j))^{2}=2j(1-j)$. We infer that $j(1-j)j=2j$; that is $j^{2}+j^{3}=2j$.
It follows that $j^{3}=0$. Hence, $J(R)$ is nil. Since $R/J(R)$ is weakly
Boolean, $R$ is strongly weakly nil clean by Theorem 2.18. Therefore, for any
$a\in R$, there exist $b\in Nil(R)$ and $e^{2}=e$ such that $eb=be$ and
$a-1=e+b$ and $a+1=e-b$. By (5), $(1+b)^{2}=1$, so $1+b$ is a tripotent. So,
$R$ is a strong weakly SIT-ring. Moreover, since $R/J(R)$ is weakly Boolean,
$2\in J(R)$ as required. ∎

######  Proposition 2.20.

If $R$ is a strong weakly SIT-ring with $3=0$ then $R$ is a subdirect product
of $\mathbb{Z}_{3}^{\prime}s$.

###### Proof.

Let $a\in R$, and write $a=e+f$ or $a=e-f$ where $e^{2}=e$, $f^{3}=f$, and
$ef=fe$. Then $a^{3}=(e+f)^{3}=e^{3}+3e^{2}f+3ef^{2}+f^{3}=e+f=a$ or
$a^{3}=(e-f)^{3}=e^{3}-3e^{2}f+3ef^{2}-f^{3}=e-f=a$. Hence, $R$ has the
identity $x^{3}=x$. Since $3=0$ in $R$, $R$ is s subdirect product of
$\mathbb{Z}_{3}^{\prime}s$. ∎

######  Proposition 2.21.

If $R$ is a strong weakly SIT-ring with $5=0$ then $R$ is a subdirect product
of $\mathbb{Z}_{5}^{\prime}s$.

###### Proof.

Let $a\in R$, and write $a=e+f$ or $a=e-f$ where $e^{2}=e$, $f^{3}=f$, and
$ef=fe$. Then
$a^{5}=(e+f)^{5}=e^{5}+5e^{4}f+10e^{3}f^{2}+10e^{2}f^{3}+5ef^{4}+f^{5}=e+f=a$
or
$a^{5}=(e-f)^{5}=e^{5}-5e^{4}f+10e^{3}f^{2}-10e^{2}f^{3}+5ef^{4}-f^{5}=e-f=a$.
Hence, $R$ has the identity $x^{5}=x$. Since $5=0$ in $R$, $R$ is s subdirect
product of $\mathbb{Z}_{5}^{\prime}s$. ∎

######  Proposition 2.22.

Every SIT-ring is a weakly SIT-ring but the converse is not true.

######  Example 2.23.

Consider the group ring $\mathbb{Z}_{6}C_{2}=\\{\varSigma_{g\in
C_{2}}a_{g}g|a_{g}\in\mathbb{Z}_{6}\\}\\\
=\\{\bar{0},\bar{1},\bar{2},\bar{3},\bar{4},\bar{5},x,\bar{2}x,\bar{3}x,\bar{4}x,\bar{5}x,\bar{1}+x,\bar{2}+x,\bar{3}+x,\bar{4}+x,\bar{5}+x,\bar{1}+\bar{2}x,\bar{2}+\bar{2}x,\bar{3}+\bar{2}x,\bar{4}+\bar{2}x,\bar{5}+\bar{2}x,\bar{1}+\bar{3}x,\bar{2}+\bar{3}x,\bar{3}+\bar{3}x,\bar{4}+\bar{3}x,\bar{5}+\bar{3}x,\bar{1}+\bar{4}x,\bar{2}+\bar{4}x,\bar{3}+\bar{4}x,\bar{4}+\bar{4}x,\bar{5}+\bar{4}x,\bar{1}+\bar{5}x,\bar{2}+\bar{5}x,\bar{3}+\bar{5}x,\bar{4}+\bar{5}x,\bar{5}+\bar{5}x\\}$.
Here,

$Id(\mathbb{Z}_{6}C_{2})=\\{\bar{0},\bar{1},\bar{2},\bar{3},\bar{4},\bar{2}+\bar{2}x,\bar{2}+\bar{4}x,\bar{5}+\bar{4}x,\bar{5}+\bar{2}x\\}$
and
$Tr(\mathbb{Z}_{6}C_{2})=\\{\bar{0},\bar{1},\bar{2},\bar{3},\bar{4},\bar{5},\\\
x,\bar{2}x,\bar{3}x,\bar{4}x,\bar{5}x,\bar{2}+x,\bar{4}+x,\bar{1}+\bar{2}x,\bar{2}+\bar{2}x,\bar{4}+\bar{2}x,\bar{5}+\bar{2}x,\bar{2}+\bar{3}x,\bar{4}+\bar{3}x,\bar{2}+\bar{4}x,\bar{3}+\bar{4}x,\bar{4}+\bar{4}x,\bar{5}+\bar{4}x,\bar{2}+\bar{5}x,\bar{4}+\bar{5}x\\}$.
Every element of $\mathbb{Z}_{6}C_{2}$ is a sum of an idempotent and a
tripotent except $\bar{5}+\bar{5}x$. But the element $\bar{5}+\bar{5}x$ is the
difference of an idempotent and a tripotent i.e
$\bar{5}+\bar{5}x=\bar{1}-(\bar{2}+x)$. Therefore, the group ring
$\mathbb{Z}_{6}C_{2}$ is a weakly SIT-ring.

######  Example 2.24.

The ring $T_{2}(\mathbb{Z}_{2})$ and $T_{2}(\mathbb{Z}_{3})$ are two weakly
SIT-ring, but not commutative.

######  Theorem 2.25.

Let $G$ be group. Then, the group ring $RG$ is semisimple if and only if the
following conditions hold.

  1. 1.

$R$ is a semisimple ring.

  2. 2.

$G$ is finite.

  3. 3.

$\mid G\mid$ is invertible in $R$.

###### Proof.

This was proved in [15]. ∎

######  Corollary 2.26.

[15], let $G$ be a finite group and let $K$ be a field. Then, $KG$ is
semisimple if and only if $char(K)\nmid\mid G\mid$.

######  Example 2.27.

The weakly SIT-ring $F_{3}C_{2}$ is semisimple, but the weakly SIT-ring
$F_{2}C_{2}$ is not semisimple by Corollary 2.26.

##  3 SIT-ring properties in Amalgamated algebras along an ideal

We begin with the following examples:

######  Example 3.1.

Let $A=\mathbb{Z}_{2}$ and $B=\begin{bmatrix}\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
0&\mathbb{Z}_{2}\end{bmatrix}$ be the rings and
$J=\begin{bmatrix}0&\mathbb{Z}_{2}\\\ 0&\mathbb{Z}_{2}\end{bmatrix}$ the ideal
of $B$ and $f:A\longrightarrow B$ defined by $f(a)=\begin{bmatrix}a&0\\\
0&a\end{bmatrix}$ where $a\in\mathbb{Z}_{2}$. Then
$f(A)+J=\Bigg{\\{}\begin{bmatrix}0&0\\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\\
0&1\end{bmatrix},\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\\
0&1\end{bmatrix},\begin{bmatrix}1&0\\\ 0&1\end{bmatrix},\begin{bmatrix}1&0\\\
0&0\end{bmatrix},\begin{bmatrix}1&1\\\ 0&1\end{bmatrix},\begin{bmatrix}1&1\\\
0&0\end{bmatrix}\Bigg{\\}}$. Hence, $A$ and $f(A)+J$ are SIT-rings. Also,
$A\bowtie^{f}J=\Bigg{\\{}(0,\begin{bmatrix}0&0\\\
0&0\end{bmatrix}),(0,\begin{bmatrix}0&0\\\
0&1\end{bmatrix}),(0,\begin{bmatrix}0&1\\\
0&0\end{bmatrix}),(0,\begin{bmatrix}0&1\\\ 0&1\end{bmatrix}),\\\
(1,\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}),(1,\begin{bmatrix}1&0\\\
0&0\end{bmatrix}),(1,\begin{bmatrix}1&1\\\
0&1\end{bmatrix}),(1,\begin{bmatrix}1&1\\\ 0&0\end{bmatrix})\Bigg{\\}}$ ,
where $Id(A\bowtie^{f}J)=\Bigg{\\{}(0,\begin{bmatrix}0&0\\\
0&0\end{bmatrix}),\\\ (0,\begin{bmatrix}0&0\\\
0&1\end{bmatrix}),(0,\begin{bmatrix}0&1\\\
0&1\end{bmatrix}),(1,\begin{bmatrix}1&0\\\
0&1\end{bmatrix}),(1,\begin{bmatrix}1&0\\\
0&0\end{bmatrix}),(1,\begin{bmatrix}1&1\\\ 0&0\end{bmatrix})\Bigg{\\}}$ and
$Tr(A\bowtie^{f}J)=\\\ \Bigg{\\{}(0,\begin{bmatrix}0&0\\\
0&0\end{bmatrix}),(0,\begin{bmatrix}0&0\\\
0&1\end{bmatrix}),(0,\begin{bmatrix}0&1\\\
0&1\end{bmatrix}),(1,\begin{bmatrix}1&0\\\
0&1\end{bmatrix}),(1,\begin{bmatrix}1&0\\\
0&0\end{bmatrix}),(1,\begin{bmatrix}1&1\\\
0&1\end{bmatrix}),(1,\begin{bmatrix}1&1\\\ 0&0\end{bmatrix})\Bigg{\\}}$ is a
SIT-ring.

######  Example 3.2.

Let $A=\mathbb{Z}_{2}$ and
$B=\begin{bmatrix}\mathbb{Z}_{2}&0&\mathbb{Z}_{2}\\\ 0&\mathbb{Z}_{2}&0\\\
0&0&\mathbb{Z}_{2}\end{bmatrix}$ be the rings and $J=\begin{bmatrix}0&0&0\\\
0&\mathbb{Z}_{2}&0\\\ 0&0&0\end{bmatrix}$ the ideal of $B$ and
$f:A\longrightarrow B$ defined by $f(a)=\begin{bmatrix}a&0&0\\\ 0&a&0\\\
0&0&a\end{bmatrix}$ where $a\in\mathbb{Z}_{2}$. Then
$f(A)+J=\Bigg{\\{}\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\par\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix}\Bigg{\\}}$. Hence, $A$ and $f(A)+J$ are SIT-rings.

Also, $A\bowtie^{f}J=\Bigg{\\{}(0,\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{bmatrix}),(0,\begin{bmatrix}0&1&0\\\ 0&1&0\\\
0&0&0\end{bmatrix}),(1,\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}),(1,\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix})\Bigg{\\}}$, where
$Id(A\bowtie^{f}J)=\Bigg{\\{}(0,\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{bmatrix}),(0,\begin{bmatrix}0&1&0\\\ 0&1&0\\\
0&0&0\end{bmatrix}),(1,\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}),(1,\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix})\Bigg{\\}}=Tr(A\bowtie^{f}J)$. Hence, $A\bowtie^{f}J$ is a
SIT-ring.

######  Definition 3.3.

A ring is called uniquely SIT-ring if each element in $R$ can be written
uniquely as the sum of an idempotent and a tripotent.

######  Proposition 3.4.

Let $f:A\rightarrow B$ be a ring homomorphism and $J$ an ideal of $B$. If
$A\bowtie^{f}J$ is a uniquely SIT-ring then $A$ is a uniquely SIT-ring.

###### Proof.

The case $J=(0)$. In view of [6, Proposition 5.1 (3)], $A$ is a SIT-ring.
Otherwise, by the same reference, the ring $A$ is a proper homomorphic image
of $A\bowtie^{f}J$. In light of Proposition 2.2, the ring $A$ is SIT-ring.
Since $A\bowtie^{f}J$ is uniquely SIT-ring and consider
$e+t=e^{\prime}+t^{\prime}$ where $e,e^{\prime}\in Id(A)$ and $t,t^{\prime}\in
Tr(A)$. Then,
$(e,f(e))+(t,f(t))=(e^{\prime},f(e^{\prime}))+(t^{\prime},f(t^{\prime}))$ and
clearly $(e,f(e)),(e^{\prime},f(e^{\prime}))\in Id(A\bowtie^{f}J)$ and
$(t,f(t)),(t^{\prime},f(t^{\prime}))\in Tr(A\bowtie^{f}J)$. Then,
$(e,f(e))=(e^{\prime},f(e^{\prime}))$ and
$(t,f(t))=(t^{\prime},f(t^{\prime}))$. Hence, $e=e^{\prime}$ and
$t=t^{\prime}$. Consequently, $A$ is uniquely SIT-ring.

∎

######  Proposition 3.5.

Let $f:A\rightarrow B$ be a ring homomorphism and $J$ an ideal of $B$. If
$A\bowtie^{f}J$ is a SIT-ring then $A$ and $f(A)+J$ are SIT-rings.

###### Proof.

The cases $J=(0)$ and $f^{-1}(J)=(0)$ follow easily from [6, Proposition 5.1
(3)]. Otherwise, by the same reference, the rings $A$ and $f(A)+J$ are proper
homomorphic images of $A\bowtie^{f}J$. By virtue of Proposition 2.2, they are
SIT-rings.

######  Proposition 3.6.

Assume that $\displaystyle\frac{f(A)+J}{J}$ is uniquely SIT-ring. Then
$A\bowtie^{f}J$ is a SIT-ring if and only if $A$ and $f(A)+J$ are SIT-rings.

###### Proof.

If $A\bowtie^{f}J$ is a SIT-ring, then so $A$ and $f(A)+J$ by Proposition 2.2.
Conversely, assume that $A$ and $f(A)+J$ are SIT-rings and consider $(a,j)\in
A\times J$. Since $A$ is SIT-ring, we can write $a=e+t$, where $e\in Id(A)$
and $t\in Tr(A)$. On the other hand, since $f(A)+J$ is SIT-ring,
$f(a)+j=f(x)+j_{1}+f(y)+j_{2}$ with $f(x)+j_{1}$ and $f(y)+j_{2}$ are
respectively an idempotent and a tripotent element of $f(A)+J$. It is clear
that $\overline{f(x)}=\overline{f(x)+j_{1}}$ (resp. $\overline{f(e)}$) and
$\overline{f(y)}=\overline{f(y)+j_{2}}$ (resp. $\overline{f(t)}$) are
respectively an idempotent and a tripotent element of
$\displaystyle\frac{f(A)+J}{J}$, and we have
$\overline{f(a)}=\overline{f(e)}+\overline{f(t)}=\overline{f(x)}+\overline{f(y)}$.
Thus, $\overline{f(e)}=\overline{f(x)}$ and $\overline{f(t)}=\overline{f(y)}$
since $\displaystyle\frac{f(A)+J}{J}$ is uniquely SIT-ring. Consider
$j^{\prime}_{1},j^{\prime}_{2}\in J$ such that $f(x)=f(e)+j^{\prime}_{1}$ and
$f(y)=f(t)+j^{\prime}_{2}$. We have,
$(a,f(a)+j)=(e,f(e)+j^{\prime}_{1}+j_{1})+(t,f(t)+j^{\prime}_{2}+j_{2})$, and
it is clear that $(e,f(e)+j^{\prime}_{1}+j_{1})$ is an idempotent and
$(t,f(t)+j^{\prime}_{2}+j_{2})$ a tripotent element of $A\bowtie^{f}J$. ∎

######  Remark 3.7.

Let $f:A\rightarrow B$ be a ring homomorphism and $J$ an ideal of $B$.

  1. 1.

If $B=J$ then, $A\bowtie^{f}B$ is SIT-ring if and only if $A$ and $B$ are SIT-
ring since $A\bowtie^{f}B=A\times B$.

  2. 2.

If $f^{-1}(J)=\\{0\\}$ then, $A\bowtie^{f}J$ is SIT-ring if and only if
$f(A)+J$ is SIT-ring (by [6, Proposition 5.1(3)]).

######  Definition 3.8.

Let $A$ and $B$ be two rings with unity. Let $J$ be an ideal of $B$ and let
$f:A\rightarrow B$ be a ring homomorphism. In this setting, we can consider
the following subring of $A\times B$:

$A\bowtie^{f}J:=\\{(a,f(a)+j)\mid a\in A,j\in J\\}$

called _the amalgamation of $A$ with $B$ along $J$ with respect to $f$_. When
$A=B$ and $f=id_{A}$, the amalgamated $A\bowtie^{id_{A}}J$ is called
amalgamated duplication of a ring $A$ along the ideal $J$, and denoted by
$A\bowtie J$.

######  Corollary 3.9.

Let $A$ be a ring and $I$ an ideal such that $A/I$ is a uniquely SIT-ring.
Then, $A\bowtie I$ is SIT-ring if and only if $A$ is a SIT-ring.

######  Proposition 3.10.

Let $f:A\rightarrow B$ be a ring homomorphism and let $(e)$ be an ideal of $B$
generated by the idempotent element $e$. Then $A\bowtie^{f}(e)$ is SIT-ring if
and only if $A$ and $f(A)+(e)$ are SIT-ring.
In particular, if $e$ is an idempotent element of $A$ then, $A\bowtie(e)$ is
SIT-ring if and only if $A$ is a SIT-ring.

###### Proof.

In light of Proposition 3.5, we have only to show that $A\bowtie^{f}(e)$ is
SIT-ring provided $A$ and $f(A)+(e)$ are SIT-ring. Let $(a,f(a)+re)$ be an
element of $A\bowtie^{f}(e)$ (with $a\in A$ and $r\in B$). Since $A$ and
$f(A)+(e)$ are SIT-ring, there exists $s$ and $t$ (resp. $s^{\prime}$ and
$t^{\prime}$) in $A$ (resp. $f(A)+(e)$) which are respectively idempotent and
tripotent such that $a=s+t$ and $f(a)+re=s^{\prime}+t^{\prime}$. We have

$(a,f(a)+re)=(s,f(s)+(s^{\prime}-f(s))e)+(t,f(t)+(t^{\prime}-f(t))e)$

On the other hand,

$\displaystyle[f(s)+(s^{\prime}-f(s))e]^{2}$ $\displaystyle=$
$\displaystyle[f(s)(1-e)+s^{\prime}e]^{2}$ $\displaystyle=$ $\displaystyle
f(s)(1-e)+s^{\prime}e$ $\displaystyle=$ $\displaystyle
f(s)+(s^{\prime}-f(s))e$

and

$\displaystyle[f(t)+(t^{\prime}-f(t))e]^{3}$ $\displaystyle=$
$\displaystyle[f(t)(1-e)+t^{\prime}e]^{3}$ $\displaystyle=$ $\displaystyle
f(t)(1-e)+t^{\prime}e$ $\displaystyle=$ $\displaystyle
f(t)+(t^{\prime}-f(t))e$

Then, $(s,f(s)+(s^{\prime}-f(s))e)$ and $(t,f(t)+(t^{\prime}-f(t))e)$ are
respectively idempotent and tripotent in $A\bowtie^{f}(e)$. Consequently,
$A\bowtie^{f}(e)$ is SIT-ring, as desired.
Finally, if $A=B$ and $f=id_{A}$ then $A\bowtie^{f}(e)=A\bowtie(e)$ and
$f(A)+(e)=A$. Thus, the particular case is a direct consequence of what is
above. ∎

######  Proposition 3.11.

Let $f:A\rightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$.

  1. 1.

If $J\subseteq Id(B)$ and $3=0$ then, $A\bowtie^{f}J$ is SIT-ring if and only
if $A$ is SIT-ring.

  2. 2.

The ring $A\bowtie^{f}J$ is tripotent if and only if $A$ is tripotent and
$J\subseteq Tr(B)$ and $3=0$ in $B$.

###### Proof.

Note first that if $J\subseteq Id(B)$ then $2J=(0)$. Indeed, let $j\in J$.
Clearly, $2j\in J\subseteq Id(B)$. Then,
$j+j=(j+j)^{2}=j^{2}+2j^{2}+j^{2}=j+2j+j$. Hence, $2j=0$.

$(1)$ Let $(a,f(a)+j)$ be an element of $A\bowtie^{f}J$ (with $a\in A$ and
$j\in J$). We have $a=e+t$ where $e$ and $t$ are respectively an idempotent
and a tripotent in $A$. We have $(f(e)+j)^{2}=f(e)^{2}+j^{2}+2f(e)j=f(e)+j$
since $2j=0$ and also $J\subseteq Tr(B)$ and $3=0$ in $B$ then, we have
$(f(t)+j)^{3}=f(t)^{3}+3f(t)^{2}j+3f(t)j^{2}+j^{3}=f(t)+j$. Hence,
$(e,f(e)+j)$ and $(t,f(t)+j)$ are respectively an idempotent and tripotent in
$A\bowtie^{f}J$, and we have $(a,f(a)+j)=((e,f(e)+j)+(t,f(t)+j)$.
Consequently, $A\bowtie^{f}J$ is SIT-ring.
The converse implication is clear.

$(2)$ If $A\bowtie^{f}J$ is tripotent, for each $a\in A$,
$(a,f(a))=(a,f(a))^{3}=(a^{3},f(a)^{3})$. Then, $a=a^{3}$ . Hence, $A$ is
tripotent. Moreover, for each $j\in J$, $(0,j)=(0,j)^{3}=(0,j^{3})$. Thus,
$j=j^{3}$. Hence, $J\subseteq Tr(B)$.
Now, assume that $A$ is tripotent and $J\subseteq Tr(B)$ and $3=0$ in $B$.
Hence, for each $a\in A$ and $j\in J$,
$(a,f(a)+j)^{3}=(a^{3},f(a)^{3}+3f(j)^{2}j+3f(a)j^{2}+j^{3})=(a,f(a)+j)$.
Thus, $A\bowtie^{f}J$ is tripotent. ∎

##  4 SITT-ring properties in bi-amalgamated algebras along ideals

The matrix ring $M_{2}(\mathbb{Z}_{2})$ is not a SIT-ring. Because every
element of $M_{2}(\mathbb{Z}_{2})$ is sum of an idempotent and a tripotent
except the element $\begin{pmatrix}\bar{0}&\bar{1}\\\
\bar{1}&\bar{0}\end{pmatrix}$ in $M_{2}(\mathbb{Z}_{2})$ is not a sum of an
idempotent and a tripotent. But, the element
$\begin{pmatrix}\bar{0}&\bar{1}\\\ \bar{1}&\bar{0}\end{pmatrix}$ can be
written as sum of an idempotent and two tripotents. In this situation, we
introduce the following definition and some properties of SITT-ring.

######  Definition 4.1.

A ring is called a (strong) SITT-ring if every element is a sum of an
idempotent and two tripotents (that commute with one another).

######  Example 4.2.

The matrix ring $M_{2}(\mathbb{Z}_{2})$ is a SITT-ring but not a SIT-ring.

######  Proposition 4.3.

The class of SITT-rings is closed under direct product and homomorphic images.

######  Proposition 4.4.

If $R$ is a strong SITT-ring and $2\in Nil(R)$, then $a^{2}-a$ is nilpotent
for all $a\in R$

###### Proof.

Let $a\in R$ and write $a=e+f+g,$ where $e\in Id(R),f,g\in Tr(R).$ Then
$a^{2}=e+f^{2}+g^{2}(mod\ 2R).$ Thus $a^{2}-a\equiv(f^{2}-f)+(g^{2}-g)(mod\
2R),$ with $(f^{2}-f)^{2}=2(f^{2}-f)$, and $(g^{2}-g)^{2}=2(g^{2}-g)$. So
$(f^{2}-f)$ and $(g^{2}-g)$ are nilpotent. Hence, $a^{2}-a$ is nilpotent.

∎

######  Proposition 4.5.

If $R$ is a strong SITT-ring and $3\in Nil(R)$, then $a^{3}-a$ is nilpotent
for all $a\in R$.

###### Proof.

Let $a\in R$ and write $a=e+f+g,$ where $e\in Id(R),f,g\in Tr(R).$ Then

$a^{3}=(e+f+g)+3ef+3eg+3ef^{2}+3f^{2}g+3eg^{2}+3fg^{2}+6efg.$

So $a^{3}-a=3ef+3eg+3ef^{2}+3f^{2}g+3eg^{2}+3fg^{2}+6efg.$ Since $3\in
Nil(R),a^{3}-a$ is nilpotent.

∎

######  Proposition 4.6.

( [13, Theorem 2.1]) Let $R$ be a ring with $J(R)=0$ such that every nonzero
right ideal contains a nonzero idempotent. If $a^{n}=0$ but $a^{n-1}\neq 0$,
then there exists $e^{2}=e\in RaR$ such that $eRe\cong\mathbb{M}_{n}(T)$ for
some nontrivial ring $T$.

######  Theorem 4.7.

If $R$ is a strong SITT-ring and $3\in Nil(R)$, then $J(R)$ is nil and
$R/J(R)$ is a subdirect product of $\mathbb{Z}_{3}$’s.

###### Proof.

By Proposition 4.5, $a^{3}-a$ is nilpotent for all $a\in R$. Then for $j\in
J(R),\ j(j^{2}-1)=j^{3}-j$ is nilpotent. So $j$ is nilpotent as $j^{2}-1\in
U(R)$. Thus, $J(R)$ is nil. For any $a\in R,\ (a-a^{3})^{n}=0$ for some $n\geq
1.$ Then $a^{n}(1-a^{2})^{n}=0$, and it follows that $a^{n}\in a^{n+1}R\cap
Ra^{n+1}$. So $R$, and hence $\bar{R}:=R/J(R)$ are strongly $\pi$-regular. We
next show that $\bar{R}$ is reduced. Assume that $\bar{a}^{2}=0$ for some
$0\neq\bar{a}\in R$. Then, by Proposition 4.6, there exists
$\bar{0}\neq\bar{y}^{2}=\bar{y}\in\bar{R}\bar{y}\bar{R}$ such that
$\bar{y}\bar{R}\bar{y}\cong\mathbb{M}_{2}(T),$ where $T$ is a nontrivial ring.
Let $s=\begin{pmatrix}1&1\\\ 1&0\end{pmatrix}\in\mathbb{M}_{2}(T).$ We have
$s^{3}=\begin{pmatrix}3&2\\\ 2&1\end{pmatrix}=\begin{pmatrix}0&2\\\
2&1\end{pmatrix}$. So $s-s^{3}=\begin{pmatrix}1&-1\\\ -1&-1\end{pmatrix}$. We
calculate that $(s-s^{3})^{2}=\begin{pmatrix}2&0\\\ 0&2\end{pmatrix}$, which
is not nilpotent. Therfore, $s-s^{3}$ is not nilpotent, a contradiction. This
shows that $\bar{R}$ is reduced. So $s^{3}=s$ for all $s\in\bar{R}$. Hence,
$\bar{R}$ is a subdirect product of $\mathbb{Z}_{3}$’s (see [14, Ex. 12.11,
p200]).

∎

######  Theorem 4.8.

Let $R$ be a strongly clean ring and $2\in Nil(R)$, If $R$ is a strong SITT-
ring, then $R/J(R)$ is Boolean with $J(R)$ is Nil.

###### Proof.

By Proposition 4.4, $a^{2}-a$ is nilpotent. Then, by [12, Theorem 2.1], $R$ is
srongly nil-clean, so $R/J(R)$ is Boolean with $J(R)$ is Nil, by [12, Theorem
2.7]. ∎

######  Definition 4.9.

If $\alpha:A\longrightarrow C$, $\beta:B\longrightarrow C$ are two ring
homomorphisms. Then the subring $D:=\alpha\times_{C}\beta:=\\{(a,b)\in A\times
B\mid\alpha(a)=\beta(b)\\}$ is called the pullback of $\alpha$ and
$\beta$.(see[6])

######  Definition 4.10.

Let $f:A\longrightarrow B$ and $g:A\longrightarrow C$ be two ring
homomorphisms and let $J$ and $J^{\prime}$ be two ideals of $B$ and $C$,
respectively, such that $I_{0}:=f^{-1}J=g^{-1}J^{\prime}$. The bi-amalgamation
of $A$ with $(B,C)$ along $(J,J^{\prime})$ with respect to $(f,g)$ is the
subring of $(B\times C)$ given by
$A\bowtie^{f,g}(J,J^{\prime}):=\\{(f(a)+j,g(a)+j^{\prime})\rvert a\in
A,(j,j^{\prime})\in J\times J^{\prime}\\}$.

Following [11], the above definition was introduced by and studied by Kabbaj,
Louartiti and Tamekkante in 2013.

######  Theorem 4.11.

Let $f:A\longrightarrow B$ and $g:A\longrightarrow C$ be ring homomorphisms
and $J$, $J^{\prime}$ be two ideals of $B$ and $C$, respectively, such that
$I:=f^{-1}(J)=g^{-1}(J^{\prime})$. Let $u:f(A)+J\longrightarrow A/I$ defined
by $u(f(a)+j)=a+I$ and $u:g(A)+J^{\prime}\longrightarrow A/I$ defind by
$v(g(a)+j^{\prime})=a+I$, where $a\in A$, $j\in J$ and $j^{\prime}\in
J^{\prime}$. Then $A\bowtie^{f,g}(J,J^{\prime})$ is the pullback of the maps
$u$ and $v$.

###### Proof.

Consider the following diagram with $\alpha(f(a)+j,g(a)+j^{\prime})=f(a)+j$
and $v(f(a)+j,g(a)+j^{\prime})=g(a)+j^{\prime}$ where
$(f(a)+j,g(a)+j^{\prime})\in A\bowtie^{f,g}(J,J^{\prime})$.

$g(A)+J^{\prime}$$A/I$$A\bowtie^{f,g}(J,J^{\prime})$$f(A)+J$$X$$\alpha$$\beta$$u$$v$$\gamma$$\delta$$\theta$

Fig. 1

Then $u\alpha=v\beta$. For if $(f(a)+j,g(a)+j^{\prime})\in
A\bowtie^{f,g}(J,J^{\prime})$, then
$u\alpha(f(a)+j,g(a)+j^{\prime})=u(f(a)+j)=a+I$ and
$v\beta(f(a)+j,g(a)+j^{\prime})=v(g(a)+j^{\prime})=a+I$. Let $X$ be any ring
and $\gamma$ and $\delta$ ring homomorphisms such that $v\delta=u\gamma$. For
any $x\in X$, set $\gamma(x)=f(a)+j\in A$ and $\delta(x)=g(a)+j^{\prime}$.
Then $v\delta(x)=v(f(a)+j)=a+I$ and $u\delta(x)=u(g(a)+j^{\prime})=a+I$.
Define $\theta:X\longrightarrow A\bowtie^{f,g}(J,J^{\prime})$ by
$\theta(x)=(f(a)+j,g(a)+j^{\prime})$. Then
$\alpha\theta(x)=\alpha(f(a)+j,g(a)+j^{\prime})=f(a)+j=\gamma(x)$ and
$\beta\theta(x)=\beta(f(a)+j,g(a)+j^{\prime})=g(a)+j^{\prime}=\delta(x)$. On
the other hand, $\theta$ is unique, for if $\zeta$ satisfies
$\alpha\zeta=\gamma$ and $\beta\zeta=\delta$, then $\zeta=\theta$ since
$\beta$ is a monomorphism. This completes the proof. ∎

######  Example 4.12.

Let $A=\mathbb{Z}_{2}$ and $B=\begin{bmatrix}\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
0&\mathbb{Z}_{2}\end{bmatrix}$ and
$C=\begin{bmatrix}\mathbb{Z}_{2}&0&\mathbb{Z}_{2}\\\ 0&\mathbb{Z}_{2}&0\\\
0&0&\mathbb{Z}_{2}\end{bmatrix}$ be the rings and
$J=\begin{bmatrix}0&\mathbb{Z}_{2}\\\ 0&0\end{bmatrix}$ the ideal of $B$ and
$J^{\prime}=\begin{bmatrix}0&0&0\\\ 0&\mathbb{Z}_{2}&0\\\ 0&0&0\end{bmatrix}$
the ideal of $C$. Let $f:A\longrightarrow B$ defined by
$f(a)=\begin{bmatrix}a&0\\\ 0&a\end{bmatrix}$ where $a\in\mathbb{Z}_{2}$ and
$g:A\longrightarrow C$ defined by $g(a)=\begin{bmatrix}a&0&0\\\ 0&a&0\\\
0&0&a\end{bmatrix}$ where $a\in\mathbb{Z}_{2}$. Then
$f(A)+J=\Bigg{\\{}\begin{bmatrix}0&0\\\ 0&0\end{bmatrix},\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\begin{bmatrix}1&1\\\
0&1\end{bmatrix}\Bigg{\\}}$ and
$g(A)+J^{\prime}=\Bigg{\\{}\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\par\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix}\Bigg{\\}}$. Hence, $f(A)+J$ and $g(A)+J^{\prime}$ are SITT-
rings. Also, $A\bowtie^{f,g}(J,J^{\prime})=\Bigg{\\{}(\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{bmatrix}),(\begin{bmatrix}0&1\\\
0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\ 0&1&0\\\ 0&0&0\end{bmatrix}),\\\
(\begin{bmatrix}0&0\\\ 0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\ 0&1&0\\\
0&0&0\end{bmatrix}),(\begin{bmatrix}0&1\\\
0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{bmatrix}),(\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}),(\begin{bmatrix}1&1\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&0&0\\\ 0&0&1\end{bmatrix}),\\\
(\begin{bmatrix}1&0\\\ 0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix}),(\begin{bmatrix}1&1\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix})\Bigg{\\}}$, where $Id(A\bowtie^{f,g}(J,J^{\prime}))=\\\
\Bigg{\\{}(\begin{bmatrix}0&0\\\ 0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\
0&0&0\\\ 0&0&0\end{bmatrix}),(\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\ 0&1&0\\\
0&0&0\end{bmatrix}),\\{(\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}),\\{(\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix})\Bigg{\\}}$ and
$Tr(A\bowtie^{f,g}(J,J^{\prime}))=\Bigg{\\{}(\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{bmatrix}),(\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\begin{bmatrix}0&0&0\\\ 0&1&0\\\
0&0&0\end{bmatrix}),(\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{bmatrix}),\\\
(\begin{bmatrix}1&0\\\ 0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix}),(\begin{bmatrix}1&1\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{bmatrix}),(\begin{bmatrix}1&1\\\
0&1\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix})\Bigg{\\}}$ is a SITT-ring.

######  Proposition 4.13.

If $A\bowtie^{f,g}(J,J^{\prime})$ is a SITT-ring then $f(A)+J$ and
$g(A)+J^{\prime}$ are a SITT-rings.

###### Proof.

Clearly, homomorphic image of a SITT-ring is a SITT-ring. Thus, in view of
[11, Proposition 4.1], we have the following isomorphism of rings
$\frac{A\bowtie^{f,g}(J,J^{\prime})}{0\times J^{\prime}}\cong f(A)+J$ and
$\frac{A\bowtie^{f,g}(J,J^{\prime})}{J\times 0}\cong g(A)+J^{\prime}$. Hence,
$f(A)+J$ and $g(A)+J^{\prime}$ are SITT-rings. ∎

######  Definition 4.14.

A ring is called uniquely SITT-ring if each element in $R$ can be written
uniquely as the sum of an idempotent and two tripotents.

######  Proposition 4.15.

Assume that $A$ is SITT-ring and $\displaystyle\frac{f(A)+J}{J}$ and
$\displaystyle\frac{g(A)+J^{\prime}}{J^{\prime}}$ are uniquely SITT-rings.
Then $A\bowtie^{f,g}(J,J^{\prime})$ is a SITT-ring if and only if $f(A)+J$ and
$g(A)+J^{\prime}$ are SITT-rings.

###### Proof.

If $A\bowtie^{f,g}(J,J^{\prime})$ is a SITT-ring, then so $f(A)+J$ and
$g(A)+J^{\prime}$ by Proposition 4.13. Conversely, assume that $f(A)+J$ and
$g(A)+J^{\prime}$ are SITT-rings. Since $A$ is SITT-ring, we can write
$a=e+t_{1}+t_{2}$, where $e\in Id(A)$ and $t_{1},t_{2}\in Tr(A)$. On the other
hand, since $f(A)+J$ is SITT-ring, $f(a)+j=f(x)+j_{1}+f(y)+j_{2}+f(z)+j_{3}$
with $f(x)+j_{1}$ and $f(y)+j_{2}$, $f(z)+j_{3}$ are respectively an
idempotent and a tripotent elements of $f(A)+J$. It is clear that
$\overline{f(x)}=\overline{f(x)+j_{1}}$ (resp. $\overline{f(e)}$) and
$\overline{f(y)}=\overline{f(y)+j_{2}}$ (resp. $\overline{f(t_{1})}$),
$\overline{f(z)}=\overline{f(z)+j_{3}}$ (resp. $\overline{f(t_{2})}$) are
respectively an idempotent and a tripotent elements of
$\displaystyle\frac{f(A)+J}{J}$, and we have
$\overline{f(a)}=\overline{f(e)}+\overline{f(t_{1})}+\overline{f(t_{2})}=\overline{f(x)}+\overline{f(y)}+\overline{f(z)}$.
Thus, $\overline{f(e)}=\overline{f(x)}$, $\overline{f(t_{1})}=\overline{f(y)}$
and $\overline{f(t_{2})}=\overline{f(z)}$ since
$\displaystyle\frac{f(A)+J}{J}$ is uniquely SITT-ring. Consider
$j^{*}_{1},j^{*}_{2},j^{*}_{3}\in J$ such that $f(x)=f(e)+j^{*}_{1}$ ,
$f(y)=f(t_{1})+j^{*}_{2}$ and $f(z)=f(t_{2})+j^{*}_{3}$ and also since
$g(A)+J^{\prime}$ is SITT-ring,
$g(a)+j^{\prime}=g(x)+j^{\prime}_{1}+g(y)+j^{\prime}_{2}+g(z)+j^{\prime}_{3}$
with $g(x)+j^{\prime}_{1}$ and $g(y)+j^{\prime}_{2}$, $g(z)+j^{\prime}_{3}$
are respectively an idempotent and a tripotent elements of $g(A)+J^{\prime}$.
It is clear that $\overline{g(x)}=\overline{g(x)+j^{\prime}_{1}}$ (resp.
$\overline{g(e)}$) and $\overline{g(y)}=\overline{g(y)+j^{\prime}_{2}}$ (resp.
$\overline{g(t_{1})}$), $\overline{g(z)}=\overline{g(z)+j^{\prime}_{3}}$
(resp. $\overline{g(t_{2})}$) are respectively an idempotent and a tripotent
elements of $\displaystyle\frac{g(A)+J^{\prime}}{J^{\prime}}$, and we have
$\overline{g(a)}=\overline{g(e)}+\overline{g(t_{1})}+\overline{g(t_{2})}=\overline{g(x)}+\overline{g(y)}+\overline{g(z)}$.
Thus, $\overline{g(e)}=\overline{g(x)}$, $\overline{g(t_{1})}=\overline{g(y)}$
and $\overline{g(t_{2})}=\overline{g(z)}$ since
$\displaystyle\frac{g(A)+J^{\prime}}{J^{\prime}}$ is uniquely SITT-ring.
Consider $j^{{}^{\prime}*}_{1},j^{{}^{\prime}*}_{2},j^{{}^{\prime}*}_{3}\in
J^{\prime}$ such that $g(x)=g(e)+j^{{}^{\prime}*}_{1}$ ,
$g(y)=g(t_{1})+j^{{}^{\prime}*}_{2}$ and $g(z)=g(t_{2})+j^{{}^{\prime}*}_{3}$.
We have,
$(f(a)+j,g(a)+j^{\prime})=(f(e)+j^{*}_{1}+j_{1},g(e)+j^{{}^{\prime}*}_{1}+j^{\prime}_{1})+(f(t_{1})+j^{*}_{2}+j_{2},g(t_{1})+j^{{}^{\prime}*}_{2}+j^{\prime}_{2})+(f(t_{2})+j^{*}_{3}+j_{3},g(t_{2})+j^{{}^{\prime}*}_{3}+j^{\prime}_{3})$,
and it is clear that
$(f(e)+j^{*}_{1}+j_{1},g(e)+j^{{}^{\prime}*}_{1}+j^{\prime}_{1})$ is an
idempotent and
$(f(t_{1})+j^{*}_{2}+j_{2},g(t_{1})+j^{{}^{\prime}*}_{2}+j^{\prime}_{2})$ and
$(f(t_{2})+j^{*}_{3}+j_{3},g(t_{2})+j^{{}^{\prime}*}_{3}+j^{\prime}_{3})$ are
tripotent elements of $A\bowtie^{f,g}(J,J^{\prime})$. ∎

######  Proposition 4.16.

Let $f:A\rightarrow B$ and $f:A\rightarrow C$ be a ring homomorphisms and let
$(e_{1})$ be an ideal of $B$ generated by the idempotent element $e_{1}$ and
$(e_{2})$ be an ideal of $C$ generated by the idempotent element $e_{2}$.
Assume that $A$ is a SITT-ring. Then $A\bowtie^{f,g}((e_{1}),(e_{2}))$ is
SITT-ring if and only if $f(A)+(e_{1})$ and $g(A)+(e_{2})$ are SITT-ring.

###### Proof.

In light of Proposition 4.13, we have only to show that
$A\bowtie^{f,g}((e_{1}),(e_{2}))$ is SITT-ring provided $f(A)+(e_{1})$ and
$g(A)+(e_{2})$ are SITT-rings. Let $(f(a)+r_{1}e_{1},g(a)+r_{2}e_{2})$ be an
element of $A\bowtie^{f,g}((e_{1}),(e_{2}))$ (with $a\in A$, $r_{1}\in B$ and
$r_{2}\in C$). Since $A$ is SITT-ring, we can write $a=s+t_{1}+t_{2}$, where
$s\in Id(A)$ and $t_{1},t_{2}\in Tr(A)$ and also since $f(A)+(e_{1})$ and
$g(A)+(e_{2})$ are SITT-rings, we can write
$f(a)+r_{1}e_{1}=s^{\prime}+t^{\prime}_{1}+t^{\prime}_{2}$, where
$s^{\prime}\in Id(f(A)+(e_{1}))$ and $t^{\prime}_{1},t^{\prime}_{2}\in
Tr(f(A)+(e_{1}))$ and
$g(a)+r_{2}e_{2}=s^{\prime\prime}+t^{\prime\prime}_{1}+t^{\prime\prime}_{2}$,
where $s^{\prime\prime}\in Id(g(A)+(e_{2}))$ and
$t^{\prime\prime}_{1},t^{\prime\prime}_{2}\in Tr(g(A)+(e_{2}))$. We have
$(f(a)+r_{1}e_{1},g(a)+r_{2}e_{2})=(f(s)+(s^{\prime}-f(s))e_{1},g(s)+(s^{\prime\prime}-g(s))e_{2})+(f(t_{1})+(t_{1}^{\prime}-f(t_{1}))e_{1},g(t_{1})+(t_{1}^{\prime\prime}-g(t_{1}))e_{2})+(f(t_{2})+(t_{2}^{\prime}-f(t_{2}))e_{1},g(t_{2})+(t_{2}^{\prime\prime}-g(t_{2}))e_{2})$.
On the other hand,

$\displaystyle[f(s)+(s^{\prime}-f(s))e_{1}]^{2}$ $\displaystyle=$
$\displaystyle[f(s)(1-e_{1})+s^{\prime}e_{1}]^{2}$ $\displaystyle=$
$\displaystyle f(s)(1-e_{1})+s^{\prime}e_{1}$ $\displaystyle=$ $\displaystyle
f(s)+(s^{\prime}-f(s))e_{1}$

$\displaystyle[f(t_{1})+(t_{1}^{\prime}-f(t_{1}))e_{1}]^{3}$ $\displaystyle=$
$\displaystyle[f(t_{1})(1-e_{1})+t_{1}^{\prime}e_{1}]^{3}$ $\displaystyle=$
$\displaystyle f(t_{1})(1-e_{1})+t_{1}^{\prime}e_{1}$ $\displaystyle=$
$\displaystyle f(t_{1})+(t_{1}^{\prime}-f(t_{1}))e_{1}$

and

$\displaystyle[f(t_{2})+(t_{2}^{\prime}-f(t_{2}))e_{1}]^{3}$ $\displaystyle=$
$\displaystyle[f(t_{2})(1-e_{1})+t_{2}^{\prime}e_{1}]^{3}$ $\displaystyle=$
$\displaystyle f(t_{2})(1-e_{1})+t_{2}^{\prime}e_{1}$ $\displaystyle=$
$\displaystyle f(t_{2})+(t_{2}^{\prime}-f(t_{2}))e_{1}$

Similarly, $g(s)+(s^{\prime\prime}-g(s))e_{2}$ is an idempotent of
$g(A)+(e_{2})$ and
$g(t_{1})+(t_{1}^{\prime\prime}-g(t_{1}))e_{2},g(t_{2})+(t_{2}^{\prime\prime}-g(t_{2}))e_{2}$
are tripotents of $g(A)+(e_{2})$.Then,
$(f(s)+(s^{\prime}-f(s))e_{1},g(s)+(s^{\prime\prime}-g(s))e_{2})$ and
$(f(t_{1})+(t_{1}^{\prime}-f(t_{1}))e_{1},g(t_{1})+(t_{1}^{\prime\prime}-g(t_{1}))e_{2})$,
$(f(t_{2})+(t_{2}^{\prime}-f(t_{2}))e_{1},g(t_{2})+(t_{2}^{\prime\prime}-g(t_{2}))e_{2})$
respectively an idempotent and tripotents in $A\bowtie^{f,g}((e_{1}),(e_{2}))$
. Consequently, $A\bowtie^{f,g}((e_{1}),(e_{2}))$ is SITT-ring, as desired. ∎

## Acknowledgements

The first author was partially supported by Periyar University Research
Fellowship (Letter No: PU/AD-3/URF/015723/2020 dated 16th November 2020). The
second author was supported by DST FIST (Letter No: SR/FST/MSI-115/2016 dated
10th November 2017).

## References

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∎

11institutetext: Anh D. Phan and Lilia M. Woods 22institutetext: Department of
Physics, University of South Florida, Tampa, Florida 33620, USA
Tel.: +1-813-974-8489
Fax: +1-813-974-5813
22email<EMAIL_ADDRESS>33institutetext: N. A. Viet 44institutetext:
Institute of Physics, 10 Daotan, Badinh, Hanoi, Vietnam

# Temperature phase transition model for the DNA-CNTs-based nanotweezers

Anh D. Phan  Lilia M. Woods and N. A. Viet

(Received: date / Accepted: date)

###### Abstract

DNA and Carbon nanotubes (CNTs) have unique physical, mechanical and
electronic properties that make them revolutionary materials for advances in
technology. In state-of-the-art applications, these physical properties can be
exploited to design a type of bio-nanorobot. In this paper, we present the
behaviors of DNA-based nanotweezers and show the capabilities of controlling
the robotic device. The theoretical calculations are based on the Peyrard-
Bishop model for DNA. Furthermore, the influence of the van der Waals force
between two CNTs on the opening and closing of nanotweezers is studied in
comparison with the stretching forces of DNA.

###### Keywords:

van der Waals interaction Carbon nanotubes DNA model

##  1 Introduction

In the past several years, researchers have made much progress in synthesizing
new materials and developing fabrication techniques necessary for nanoscaled
device production. This progress has been particularly important for
applications utilizing physical systems intended for biological and medical
purposes.In this regard, biophysical devices at the nanoscale open up novel
possibilities for diagnostic and therapeutic applications.

DNA and carbon nanotubes (CNTs) are interesting and important systems in
nanoscience. They have been the subject of many investigations in the past two
decades1 ; 2 ; 3 ; 4 ; 23 . DNA is composed of two long polymer strands
organized in a double helical structure, where each strand consists of
repeating units (nucleotides)24 . CNTs are quasi-one dimensional cylindrically
wrapped graphene sheets with properties uniquely defined by theregistry
dependence of the wrapping given by a chirality index $(n,m)$ 25 . Various
applications of DNA/CNT complexes have been exploited with potential for
biosensors 6 , DNA transporters 7 , and field effect transistors 8 . The
DNA/CNT is a composite with complicated structure with temperature dependent
motion dynamics. Recently, using molecular dynamics simulations researchers
have proposed molecular tweezers combining DNA and CNTs 19  \- a device with
further technological and scientific potential.

A theoretical model of a geometrical soliton of DNA structure was constructed
for the first time by Englander 9  (E model). In this model, one of the
strands of the DNA is represented as a chain of pendula interacting with the
another fixed similar strand. The E model explains the existence of DNA open
state due to nonlinear excitations. In addition, the DNA structure and
dyanmics has been modeled in terms of the Peyrard-Bishop (PB) model 1 , which
has been succssessful in explaining DNA denaturation transitions, pre-melting
dynamics, and thermal transport. In the PB model, backbone of DNA is described
as chains of particles with nearest neighboring potentials. However, the
models ignore the helicoidal structure of the DNA molecule, the context of DNA
flexibility, and the properties associated with it.

CNTs are chemically inert and they interact with other materials via long-
ranged dispersive forces, such as van der Waals (vdW) forces. The vdW
interactions of graphitic nanostructures can be described via pairwise
interatomic Lennard-Jones (LJ) potentials 11 . This approach relies on
knowledge of the coupling Hamaker constants and it predicts the equilibrium
separation correctly. The LJ potential has been applied to model mutual
interaction between various CNTs as well as CNT based devices 26 ; 27 .

In this work we investigate the dynamics of hybrid DNA/CNT nanotweezers by
employing the PB and vdW-LJ models. This dynamics of stretching in terms of
its velocity and acceleartion due to environmental temperature changes is
investigated. The critical temperature where a melting transition of the
DNA/CNT takes place is presented. Comparisons between the strength of the
involved forces showing the temperature-dependent motion is dominated by the
stretching of the H bonds and bases, while the CNT vdW interaction is weaker.

The rest of the paper is organized as follows: In Sec. II, the theoretical
structure model, behavior and interactions of DNA-based nanotweezer are
introduced. In Sec. III, numerical results are presented. The conclusions are
given in Sec. IV.

##  2 Model and mathematical background

The proposed nanotweezer architecture is assembled by attaching the reactive
ends of two single wall CNTs to the DNA strands as shown in Fig. 1. The rest
of the end C bonds are saturated via H atoms. The size of this hybrid is quite
large, approximately thousands atoms, thus full quantum mechanical atomistic
treatment is not possible. The PB model is relatively simple 1 , which
describes the DNA two strands as a coupled pendulum system.

Figure 1: (Color online) Schematics of the DNA/CNTs-based nanotweezer.

###  2.1 Model of DNA dynamics

According to the PB model 1 , the DNA double strand is modeled by two parallel
chains of nucleotides via nearest-neighbor harmonic oscillator interactions.
The potential for the Hydrogen bonds is also included. The relevant
Hamiltonian is given as follows 1 ; 2

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\left[\frac{1}{2}m\left(\dot{u}_{n}^{2}+\dot{v}_{n}^{2}\right)+\frac{1}{2}k\left(u_{n}-u_{n-1}\right)^{2}+\right.$
(1)
$\displaystyle\left.\frac{1}{2}k\left(v_{n}-v_{n-1}\right)^{2}+V\left(u_{n}-v_{n}\right)\right],$

where $u_{n}$ and $v_{n}$ are the nucleotide displacements from equilibrium
along the direction of the hydrogen bonds for each strand. $m$ is the mass of
each nucleotide (taken to be the same for each unit), while $k$ is harmonic
oscillator coupling constant of the nearest-neighbor longitudinal interaction
along each strand in units of $eV/\AA^{2}$. The potential for the Hydrogen
bonds between the two strands is modeled via a Morse potential
$V(r)=D[e^{-\alpha r}-1]^{2}$. Here, $D$ is the dissociation energy and
$\alpha$ is a parameter. It is important to note that the Morse potential
represents the hydrogen bonds between complementary bases, the repulsive
interactions of the phosphate, and the influence of the solvent environment.

The dynamics of the system described by Eq.(1) is conveniently described using
a set of new variables $x_{n}=(u_{n}+v_{n})/\sqrt{2}$ and
$y_{n}=(u_{n}-v_{n})/\sqrt{2}$, representing the in-phase and out-of-phase
motion of the two strands, respectively. Using this separation of variables,
the Hamiltonian is decoupled. An important point is that $y_{n}$ represents
the relative displacements between two nucleoid at the site $n$ in different
strands. It reflects the stretching of DNA. Here we consider the out-of-phase
displacements stretch of the hydrogen bonds given by $H_{y}$

$\displaystyle
H_{y}=\sum_{n=1}^{N}\left[\frac{1}{2}m\dot{y}_{n}^{2}+\frac{1}{2}k\left(y_{n}-y_{n-1}\right)^{2}+V\left(2y_{n}\right)\right].$
(2)

In the case of large number of nucleotides $N\rightarrow\infty$ and $H$ is
independent on the particular site $n$. Perfroming statistical averaging in
the canonical ensemble, the Schrodinger equation of a single mode $y$ using
$H_{y}$ is given by 1 ; 15 ; 16

$\displaystyle\left(-\frac{1}{2\beta^{2}k}\frac{\partial^{2}}{\partial
y^{2}}+V(2y)\right)\varphi(y)=\varepsilon\varphi(y),$ (3)

where, $\beta=1/k_{B}T$, and $k_{B}$ is the Boltzmann constant. The exact
solution for eigenenergies is given 26

$\displaystyle\varepsilon_{n}=\frac{1}{2\beta}\ln\left(\frac{\beta
k}{2\pi}\right)+\frac{2\alpha}{\beta}\sqrt{\frac{D}{k}}\left(n+\frac{1}{2}\right)-\frac{\alpha^{2}}{\beta^{2}k}\left(n+\frac{1}{2}\right)^{2}.$
(4)

Eq.(4) has a discrete energy spectrum when $d=(\beta/\alpha)\sqrt{kD}>1/2$.
This allows one to obatin a critical temperature $T_{c}=2\sqrt{kD}/(\alpha
k_{B})$, which is considered as the melting temperature of DNA. The DNA states
are continuous for $T>T_{c}$ and discrete for $T<T_{c}$. For the parameters of
DNA, when we consider $T>200$ $K$, only the value of $n=0$ is taken into
account. There is no excitation state for DNA in our considerations.

From this, the ground state eigenfunction and eigenenergy in the
thermodynamics limit of a large system is obtained as 1 ; 2

$\displaystyle\varphi_{0}(y)=\sqrt{\sqrt{2}\alpha}\frac{(2d)^{d-1/2}}{\sqrt{\Gamma(2d-1)}}e^{-de^{-\sqrt{2}\alpha
y}}e^{-(d-1/2)\sqrt{2}\alpha y},$ (5)
$\displaystyle\varepsilon_{0}=\frac{1}{2\beta}\ln\left(\frac{\beta
k}{2\pi}\right)+\frac{\alpha}{\beta}\sqrt{\frac{D}{k}}-\frac{\alpha^{2}}{4\beta^{2}k}.$
(6)

In addition, the system described via Eq.(3) can be represented as a
quasiparticle with a tempereture dependent effective mass
$m^{*}=\hbar^{2}\beta^{2}k$. At room temperature, the value of the effective
mass is approximately $22.87$ $m_{0}$, here $m_{0}$ is the rest mass of
electron. The average stretching of the hydrogen bonds can also be calculated
via $\left\langle y\right\rangle=\int\varphi_{0}^{2}(y)ydy$ 1 ; 2 .

The stretching force is determined via the expression

$\displaystyle F_{s}=-\frac{\partial V(\left\langle
y\right\rangle)}{\partial\left\langle y\right\rangle}.$ (7)

To investigate thermal properties of DNA, we heated up and cooled down
temperature of the bio-systems flollowing an expression $T=1.14t+300$ (K) 19 .
Here $T$ (K) is the environment temperature, $t$ (ps) is time. Basing on the
average stretching $\left\langle y\right\rangle$ of the coupling constants
pointed out above, the velocity $v=d\left\langle y\right\rangle/dt$ and
acceleration $a=d^{2}\left\langle y\right\rangle/dt^{2}$ of the opening of the
nanotweezers obtained by taking the first and second derivative of the
stretching with respect to time, respectively, are presented in Fig. 2

Figure 2: (Color online) (Color online) The time-dependent velocity and
acceleration of the opening.

For $k=2.10^{-3}$ eV/$\AA^{2}$, the velocity of the opening increases and
reaches to the maximum with the value of $10.34$ m/s at around $t=20$ s. After
that, the velocity drops significantly to zero. It refers that the temperature
corresponding to the peak is $322.5$ K. On the other hand, initially, the
value of acceleration is positive and rises to the maximum value $0.187\times
10^{12}$ $m/s^{2}$ at $-5.2$ s or $294$ K before declining gradually to the
negative side, crossing the time axis at $19.4$ s or $322.5$ K, touching the
bottom $-0.164\times 10^{12}$ $m/s^{2}$ at around $45$ s and continuing to
approach to $0$. It can be easily explained due to the fact that below $322.5$
K, the stretching velocity climbs significantly, so the acceleration is
positive. Zero acceleration, of course, is at the relevant bending point of
the opening velocity. Above $322.5$ K, the unzipping velocity declines
notably, and is nearly unchanged. Therefore, the acceleration has the negative
values and goes to zero.

In the same way, for other values of $k=3.10^{-3}$ eV/$\AA^{2}$ and
$k=4.10^{-3}$ eV/$\AA^{2}$, the zero acceleration takes place at $88$ $s$ and
$138$ $s$, respectively. It means that the melting temperatures corresponds to
$401$ $K$ for $k=3.10^{-3}$ eV/$\AA^{2}$ and $456$ $K$ for $k=4.10^{-3}$
eV/$\AA^{2}$. As a result, there is a possibility to obtain the melting
temperature by observing the velocity of stretching.

###  2.2 CNT van der Waals interaction

The vdW interaction between the CNT parts of the DNA nanotweezers is described
via the Lennard-Jones (LJ) approximation. This approach is widely used in
calculating disperssive interactions between graphitic nanostructures because
of its relative simplicity and satisfactory results in determining their
equilibrium configurations 11 . The LJ potential is essentially a pairwise
apprximation, and for extended systems, one typically perfroms integration
over the volumes of the interacting objects. For CNTs, the integration is over
the surfaces of hollow cylinders with radii corresponding to the radii of the
nanotubes. The LJ-vdW potential per unit length for two parallel CNTs with
radii $R_{1}$ and $R_{2}$ is given by 10

Figure 3: (Color online) Sketch of van der Waals interaction between two CNTs.

$\displaystyle
V_{vdW}=\sigma^{2}\int\int\left(-\frac{A}{\rho^{6}}+\frac{B}{\rho^{12}}\right)dS_{1}dS_{2},$
(8)

where $A$ and $B$ are the Hamaker constants corresponding to the attractive
and repulsive contributions, respectively. For graphitic systems, one
typically takes the values for graphite $A=15.2$ eV$\AA^{6}$ and $B=24\times
10^{3}$ eV$\AA^{12}$ 11 . $\sigma=4/\sqrt{3}a^{2}$ is the mean surface density
of Carbon atoms with $a=2.49$ $\AA$ being the lattice constant. Also, the
distance between the CNT surfaces is $\rho$. Perfroming the integration over
the length of the two CNTs with radii $R_{1}$ and $R_{2}$, the LJ-vdW
interaction can be written as 10 :

$\displaystyle V_{vdW}$ $\displaystyle=$ $\displaystyle-\frac{3\pi
A\sigma^{2}R_{1}R_{2}}{8}\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{1}{r^{5}}d\varphi_{1}d\varphi_{2}$
(9) $\displaystyle+\frac{63\pi
B\sigma^{2}R_{1}R_{2}}{256}\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{1}{r^{11}}d\varphi_{1}d\varphi_{2},$

where the in-plane distance between two surface elements is defined as
$r^{2}=(R-R_{1}\cos\varphi_{1}+R_{2}\cos\varphi_{2})^{2}+(R_{1}\sin\varphi_{1}-R_{2}\sin\varphi_{2})^{2}$.
The definitions of $R_{1}$, $R_{2}$, $\varphi_{1}$, $\varphi_{2}$, and $r$ are
sketched in Fig. 3.

Then, applying the first derivative with respect to $R$, we obtain the van der
Waals interaction force per unit length

$\displaystyle F_{vdW}(R)=-\frac{\partial V_{vdW}}{\partial R}.$ (10)

##  3 Numerical results and discussions

As a prototype, we take that both CNTs are identical with the chiral vector
$(5,0)$ and $(6,0)$, and lengths $L_{1}=L_{2}=5$ $nm$. The total Hamiltonian
for the system is composed of two term, that account for the stretching and
van der Waals interaction - $H=H_{y}+V_{vdW}$. Because of the relatively weak
vdW force between the tubes, $V_{vdW}$ is treated as a perturbation compared
to $H_{y}$. The parameters of DNA are $D=0.33$ $eV$ and $\alpha=18$ $nm^{-1}$.
It is important to note that $\varphi_{0}(y)$ and $\varepsilon_{0}$ in the
previous section is the wave function and energy of the ground state of DNA
without the presence of CNTs.

In Fig. 4 and Fig. 5, we show results for the CNT vdW perturbative force
correction as a function of tempertaure and the stretching force. Fig. 5
indicates that $F_{s}$ decreases as $T$ increases. The stretching force goes
to zero at the critical temperature since the properties of DNA change when
$T$ reaches to $T_{c}$.

Figure 4: (Color online) The first-order energy is caused by the van der Waals
interactions between two CNTs.

Obviously, the wave function is temperature-dependent, so the energy and
energy shift are functions of temperature. The value of $\varepsilon_{0}$ for
three values of $k$ at this range of temperature varies from $220$ meV to
$280$ meV. It means that the influence of the van der Waals interaction on the
wave function and the energy in the ground state is minor. We can calculate
separately the interactions of DNA and CNTs. An additional point is that the
larger temperature is, the smaller the first-order pertubation of energy is. A
simple reason for this problem is that when temperature increases, two DNA
strands are opened 1  and it leads to a rapid growth of distance between two
CNTs.

It is remarkable that we have studied the van der Waals interaction and the
pertubation energy between two parallel CNTs. This configuration also is used
in order to calculate all of the van der Waals interactions below.
Nevertheless, in actual cases, we have two crossed CNTs. The dispersion
interaction in real biosystems is weaker than that in the parallel state.
Therefore, we can utilize the wave function $\varphi_{0}(y)$ in the following
calculations without addional terms due to the perturbation theory.

Figure 5: (Color online) The unzipping force as a function of temperature .

It is clearly seen in Fig. 5, at the critical temperature $T_{c}$, the
stretching force vanishes because two strands of DNA are broken for $T>T_{c}$.
The opening force of DNA is very large at low temperature. The smaller the
temperature is, the smaller distance between two strands is. This force
decreases when increasing temperature since the separation distance is larger
and larger.

These results have aggrement about the range of magnitude force with
experimental data and previous calculations 21 ; 22 . The increase of $k$
causes to the growth of stretching force due to the fact that the binding of
DNA rises.

Lets consider the interaction between two CNTs attached in the ends of DNA.
There are several types of DNA existing in nature such as B-DNA and Z-DNA.
Since the diameter of DNA is approximately $2.37$ nm for B-DNA and $1.84$ nm
for Z-DNA. We assume that the initial distance between two centers of CNTs is
$1.5$ nm. It is important to note that the van der Waals force is attractive
at this range of distance and the sign of this force should be minus. The
magnitude of van der Waals interaction between two CNTs is presented in Fig.
6.

Figure 6: (Color online) The van der Waals forces between two parallel CNTs
(5,0) and (6,0) as a function of the separation distance between two centers
of CNTs.

For $k=2.10^{-3}$ eV/$\AA^{2}$, if $T<277$ $K$, the stretching force is much
larger than the van der Waals force of CNTs (5,0) and (6,0) at the initial
state. Therefore, it is easy to control the opening and closing of DNA by
cooling down or heating up. At low temperature, the contribution of the
dispersion force in the movement of DNA strands is minor. However, it can rise
to significant role when $T>277$ $K$. We can do the same way with
$k=3.10^{-3}$ eV/$\AA^{2}$ and $k=4.10^{-3}$ eV/$\AA^{2}$.

Figure 7 shows the forces between CNT (5,0) and different CNTs at the certain
distances. In order to control the opening and closing of nanotweezers, the
van der Waals force is weaker than the stretching forces. It is difficult to
operate the movement of nanotweezers if two CNTs have large radii.

Figure 7: (Color online) The van der Waals forces between CNT (5,0) and
another CNT.

When we heat up the biosystem, two ends of DNA are separated by the stretching
force. At the larger temperature, the unzipping force is much larger than the
van der Waals interactions, the nanotweezers are opened. The obtained results
agree with the previous simulation study 19 . Therefore, in our nanorobots,
the movements of CNTs can be controlled by changing temperature. In addition,
the van der Waals interaction between two cylinders is proportional to the
length of tubes. If we want to have the smaller van der Waals interaction, it
is possible to choose the length 1 nm or 2 nm. Another point is that long CNTs
are bent because of the van der Waals interaction. As a consequence, the
length of tubes should not be large in designing the bio-nanorobots.

##  4 Conclusions

The use of intelligence, sensing and actuation nanodevices in surgery, medical
treatments and materials science is a reality which has become a hot topic in
the biomedical industry and research in recent years. Bio-nanorobots provide
further advance not only in the nanotechnology, but also efficient approaches
for disease treatment. Our studies showed the behavior and architecture of the
bio-nanotweezers. The temperature dependence of the opening displacements of
tweezers is presented and gives researchers some principles to understand the
operation of DNA-based molecular machines and devices. In addition, the
velocity and acceleration of the opening and closing tweezers as a function of
time are speculated. The theoretical calculations are easy to understand and
agree qualitatively with the previous works. Further research on these systems
can considerably extend interdisciplinary implications for the technology.

###### Acknowledgements.

We thank Professor M. Peyrard for helpful discussions and comments. We
gratefully acknowledge support through the Department of Energy under Contract
No. DE-FG02-06ER46297. The work was partly funded by the Nafosted Grant No.
103.06-2011.51.

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