Let $R$ be a ring with unity, and $Z(R_R)$ be its right singular ideal, i.e. the set of elements of $R$ whose right annihilators are essential in the right module $R_R$.
My question:
If $x\in Z(R_R)$, is there an element $y\in R$ with $y+x+yx=0$?
It is easy to see that when $1+x$ is a unit in $R$ ( for example, when $x\in Z(R_R)$ is nilpotent, which is the case whenever $R$ has the ascending chain condition on right annihilators of elements ) such a $y$ exists.
Thanks for any suggestion or help!