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.