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]. A ring $R$ is called strongly invo-regular ring if $a^2=au$for each $a\in R$and some $u\in R$with $u^2=1$[10]. We now provide our observations and results in the next section.

2. Some Important Observations and Results

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

$Inv\left(R\right)=\left\{\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\hspace{0.17em}\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right),\hspace{0.17em}\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right),\hspace{0.17em}\left(\begin{array}{cc}1& 2\\ 0& 2\end{array}\right)\right\}$. 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 a strongly invo-regular ring iff it is a tripotent ring.

Proof. Let $R$is a weakly tripotent and strongly invo-regular ring. Then $R$ is a subdirect product of copies of the field of order two and the field of order three [10]. 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 [10] it is a strongly invo-regular ring.

Corollary2.4. 

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

Corollary2.5. 

There does not exist a noncommutative strongly invo-regular ring.

Proof. 

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

Note 2.6.

Theorem 2.6 [1] states that$R$ is invo-regular ring is equivalent to $R$ is a subdirect product of the field of order two and the field of order three. In order to prove this equivalence it was used that $R$ is an invo-regular ring iff$R\cong R_1\times R_2$, where $R_1$ is an invo-regular ring of characteristic two and $R_2$ is an invo-regular ring of characteristic three and further it was concluded that $R_1$ is a Boolean ring of characteristic two. However in the view of Propositions 2.1 and 2.2 described above, the proof of this equivalence described in [1] is not necessarily valid. In case Theorem 2.6 [1] is valid then we must have the following results.

Proposition 2.7.

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

Corollary 2.8.

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

Corollary 2.9.

There does not exist a noncommutative invo-regular ring.