A Note on Invo-Regular Unital Rings

1. Introduction

In this paper each ring is a unital and associative ring and following [1] we assume that the identity element of a ring is different from the zero element. A ring R is called invo-regular if for each $a\in R$ there exists $b\in Inv\left(R\right)$ such that $a=aba$ [1,2,3]. Here $Inv\left(R\right)$ is the set of all involutions. One may note that an element $b$ of $R$ satisfying $b^2=1$ is called an involution [1,2,3] and the notion of invo-regular rings is a generalization of the well known notion of unit regular rings [4,5,6].

It should be emphasized that as per the existing literature [1, Proposition 2.5] a ring $R$ is invo-regular iff $R\cong R_1\times R_2$, here $R_1$ is an invo-regular ring of characteristic two and $R_2$ is an invo-regular ring of characteristic three.

However we prove that if $R$ is an invo-regular ring and $R\cong R_1\times R_2$, then the characteristic of $R_1$ need not be two. In addition we exhibit that if $R$ is an invo-regular ring and $R\cong R_1\times R_2$, then $R_1$ need not be Boolean. However it was asserted in [1, Proof of Theorem 2.6] that if $R$ is an invo-regular ring then $R\cong R_1\times R_2$ and $R_1$ is a ring of characteristic two which must be a Boolean ring. One may note that a ring $R$ is called Boolean if for each $a\in R$, we have the identity $a^2=a$ [7]. A ring $R$ is called tripotent if for each $a\in R$, we have the identity $a^3=a$ and a ring $R$ is called weakly tripotent if for each $a\in R$, we have the identity $a^3=a$ or ${\left(1-a\right)}^3=1-a$ [7,8].

2. Some Important Observations

Proposition 2.1: 

If $R$ is an invo-regular ring and $R\cong R_1\times R_2$, then the characteristic of $R_1$ need not be two.

Proof. 

Let $R=\left\{\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right),\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right),\left(\begin{array}{cc}2& 2\\ 0& 0\end{array}\right),\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right),\left(\begin{array}{cc}1& 2\\ 0& 2\end{array}\right),\left(\begin{array}{cc}0& 2\\ 0& 1\end{array}\right),\left(\begin{array}{cc}0& 1\\ 0& 2\end{array}\right)\right\}$.

Clearly $R$ is a commutative ring of characteristic three under addition and multiplication of matrices modulo three. We have

$Inv\left(R\right)=\left\{\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right),\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right),\left(\begin{array}{cc}1& 2\\ 0& 2\end{array}\right)\right\}$. It is easy to check that $R$ is an invo-regular unital ring. Now we have the following cases.

Case I: $R\cong R\times \left\{0\right\}$. One may note that $R$ is not a ring of characteristic two.

Case II:$R\cong \left\{0\right\}\times R$. It is clear that $\left\{0\right\}$ is not a ring of characteristic two.

Case III: $R\cong R_1\times R_2$. Here $R_1=Z_3=R_2$. We note that the characteristic of $R_1=Z_3$ is not two.

Further we emphasize that if the characteristic of $R_1$ is two, then the order of $R$ must be even. But the order of $R$ is nine. Thus we see that in the above example the characteristic of $R_1$ can never be two even though $R$ is an invo-regular ring.

Proposition 2.2: 

If $R$ is an invo-regular ring such that $R\cong R_1\times R_2$, then $R_1$ need not be a non-zero Boolean ring.

Proof. 

Let $R$ is an invo-regular ring and $R\cong R_1\times R_2$. Clearly the characteristic of $R_1$ need not be two (we refer Proposition1). But it is well known that a non-zero Boolean ring must have characteristic two, hence $R_1$ need not be a non-zero Boolean ring.

Proposition 2.3: 

A weakly tripotent ring is an invo-regular ring iff it is a tripotent ring.

Proof. 

Let $R$ is a weakly tripotent invo-regular ring. Then $R$ is a subdirect product of copies of field of order two and the field of order three [1]. Hence by [9] $R$ is tripotent. Conversely let $R$ is tripotent. Then clearly it is weakly tripotent and by [9] it is a subdirect product of copies of the field of order two and the field of order three. Therefore by [1] it is an invo-regular ring.

Corollary 2.4: 

Every invo-regular ring is a tripotent ring. The converse is also true.

Corollary 2.5: 

There does not exist a noncommutative invo-regular ring.

Proof. 

Every tripotent ring is commutative [7]. Therefore it follows from Corollary 2.4 that every invo-regular ring is commutative. Hence there does not exist a noncommutative invo-regular ring.