Classification of nonsemicommutative rings (Q2236640)

The authors determine the relationships among seven classes of rings. Let \(R\) be a unitary ring and \(N(R)\) its set of nilpotent elements. \(R\) is semicommutative (SC) if \(ab=0\) implies \(aRb=0\); 2-primal (\(2\)-\(P\)) if \(N(R)\) is the lower nil-radical; \(PS\)-\(1\) if \(R/\text{Ann}(a)\) is \(2\)-\(P\) for all \(a\); weakly 2-primal (\(W 2\)-\(P\)) if \(N(R)\) is the Levitzki radical; \(NI\) if \(N(R)\) is the upper nil radical; \(NR\) if \(N(R)\) is a non-unitary subring of \(R\); and Dedekind finite \((DF)\) if \(ab=1\) implies \(ba=1\). It is known that \[SC\subseteq PS\text{-}1\subseteq 2\text{-}P\subseteq W 2\text{-}P\subseteq NI\subseteq NR\subseteq DF.\] In this paper, the authors show by ingenious examples that all the inclusions are strict. They find examples in which \(R\) is reflexive (\(aRb=0\) implies \(bRa=0\)) or abelian (all idempotents are central).