Let $R$ be an associative and non-unital ring. (Suppose that $R$ is $s$-unital, i.e. for each $x\in R$ there is $u,v\in R$ such that $ux=xv=x$.)


It is not difficult to show that if $R$ is a simple ring, then $Z(R)=\{ 0 \}$. Thus, non-unital simple rings are in some sense "extremely" non-commutative.


Are there any (common) examples of rings satisfying the following two conditions?


(1) $Z(R)=0$


and


(2) $R$ is non-simple.



Non-simple and non-unital rings with trivial centres
Take the semigroup ring $\mathbb{Z}S$ where $S=${ $a,b,c$} with multiplication $aS=bS=a, cS=c$. The elemwents $a$ and $c$ generate an ideal.


