Problem

Source: Romanian National Olympiad 2016, grade 12, problem 2

Tags: abstract algebra, Ring Theory



Let $A$ be a ring and let $D$ be the set of its non-invertible elements. If $a^2=0$ for any $a \in D,$ prove that: a) $axa=0$ for all $a \in D$ and $x \in A$; b) if $D$ is a finite set with at least two elements, then there is $a \in D,$ $a \neq 0,$ such that $ab=ba=0,$ for every $b \in D.$ Ioan Băetu