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I am just studying s-unital rings.

A ring is called left (resp. right) s-unital if $x\in Rx$ (resp. $x\in xR$) for all $x$ in $R$. A ring is called s-unital if and only if $x\in xR\cap Rx$ for all $x$.

Do you know any books which expand this subject?

I have an example of right s-unital: $R=\left\{ \begin{pmatrix}0 & 0\\ 0 & 0 \end{pmatrix},\begin{pmatrix}1 & 1\\ 0 & 0 \end{pmatrix},\begin{pmatrix}0 & 0\\ 1 & 1 \end{pmatrix},\begin{pmatrix}1 & 1\\ 1 & 1 \end{pmatrix}\right\}$

So if I take an element $x$ from this ring and multiply by another element it should be equal to $x$. Do I understand this definition properly? However, if I multiply $\begin{pmatrix}1 & 1\\ 0 & 0 \end{pmatrix}\begin{pmatrix}0 & 0\\ 0 & 0 \end{pmatrix}$ then is not equal to \begin{pmatrix}1 & 1\\ 0 & 0 \end{pmatrix}

Could you give me another example of s-unital ring which is at the same time rigt and left s-unital?

Thank you.

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Note that $x\in xR$ doesn't imply that for all $y\in R$ it holds that $xy=x$, but rather that there exists some $y\in R$ such that $xy=x$. Also, did you intend to take $R$ as a subring of $\mathbb{F}_2^{2\times 2}$? – Jonathan Y. Aug 14 '13 at 11:47
@Jonathan Y.: Yes, I forgot to write it. – Monika Aug 14 '13 at 11:50
And I guess you are mainly interested in examples of rings without identity (since every ring with identity obviously has this property.) – rschwieb Aug 14 '13 at 12:49

(Omitting the obvious suggestion of rings with identity since it's too obvious, and since the OP's example suggests that they are thinking of rngs.)

Any boolean rng (meaning possibly without identity) is clearly left-right $s$-unital.

For a concrete example, consider this boolean ring which lacks an identity: $\oplus _{i\in \Bbb N}\Bbb F_2$.

I haven't been able to find any books with the topic, but it's clear that there are plenty of research papers appearing in journals and proceedings that would serve the same purpose.

Just go to googlebooks and googlescholar and search (in quotes) "s-unital ring," and you get lots of sources.

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