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 a2=0 for any aD, prove that: a) axa=0 for all aD and xA; b) if D is a finite set with at least two elements, then there is aD, a0, such that ab=ba=0, for every bD. Ioan Băetu