On idempotents in relation with regularity. (Q2794882)

Rings \(R\) with certain conditions for existence of idempotents are investigated. The right attaching-idempotent condition requires that for each \(a\in R\), there is a nonzero element \(b\in R\) such that \(ab\) is idempotent, while \(R\) is called generalized regular if for each nonzero \(a\in R\), there is some \(b\in R\) such that \(ab\) is a nonzero idempotent. Various characterizations, consequences, and properties of these conditions are developed.NEWLINENEWLINE For example, generalized regularity is left-right symmetric, Morita invariant, and closed under extensions of rings by ideals. The first and third properties fail for right attaching-idempotent, but that condition is Morita invariant and is preserved in formal triangular matrix rings. A strong version of generalized regularity is introduced, and it is proved that \(R\) is strongly generalized regular if and only if it is both generalized regular and abelian (all idempotents central).