Structure of idempotents in rings without identity. (Q2877780)

A ring \(R\) is called Abelian if all idempotents are central, and \(R\) is said to have the insertion-of-factors property if \(a,b\in R\) and \(ab=0\) implies \(aRb=\{0\}\). The authors generalize these notions as follows: (i) \(R\) has the right (resp. left) insertion-of-idempotents property (IIP) if whenever \(e\) is idempotent and \(abe=0\) (resp. \(eab=0\)), \(aeb=0\); (ii) \(R\) is right (resp. left) idempotent-reversible (IR) if whenever \(e\) is idempotent and \(ae=0\), then \(ea=0\) (resp. \(ea=0\), then \(ae=0\)).NEWLINENEWLINE The paper studies relationships among these properties, showing among other results that (a) IIP is not left-right symmetric; (b) IR is not left-right symmetric; (c) \(R\) is Abelian if and only if \(R\) is both left and right IIP; (d) \(R\) is Abelian if and only if \(R\) is both left and right IR: (e) right (resp. left) IIP implies right (resp. left) IR, but the converse is not true. The final section identifies all noncommutative Abelian \(R\) and noncommutative Abelian \(R\) with 1 of minimal order.