Question 136
Let $R$ be a ring with only finitely many invertible elements. Prove that the two statements below are equivalent:
(a) For every non-invertible element $x$ of $R$, there exists a non-invertible element $y$ of $R$ such that $xy = x + y$.
(b) Every non-invertible element of $R$ is nilpotent.
(An element $z$ of $R$ is called *nilpotent* if $z^k = 0$ for some positive integer $k$.)
(a) For every non-invertible element $x$ of $R$, there exists a non-invertible element $y$ of $R$ such that $xy = x + y$.
(b) Every non-invertible element of $R$ is nilpotent.
(An element $z$ of $R$ is called *nilpotent* if $z^k = 0$ for some positive integer $k$.)