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Romanian National Olympiad 2016 (correction)

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

December 29, 2023
abstract algebraRing Theory

Problem Statement

Let AA be a ring and let DD be the set of its non-invertible elements. If a2=0a^2=0 for any a∈D,a \in D, prove that: a) axa=0axa=0 for all a∈Da \in D and x∈Ax \in A; b) if DD is a finite set with at least two elements, then there is a∈D,a \in D, a≠0,a \neq 0, such that ab=ba=0,ab=ba=0, for every b∈D.b \in D.
Ioan Băetu
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