Weakly normal rings. (Q2892194)

A ring \(R\) is called weakly normal if for all \(a,r\in R\) and \(e\) in \(E(R)\), \(ae=0\) implies \(Rera\) is a nil left ideal of \(R\), where \(E(R)\) is the set of all idempotent elements of \(R\). In this paper the authors prove that \(R\) is weakly normal if and only if \(Rer(1-e)\) is a nil left ideal of \(R\) for each \(e\) in \(E(R)\) and \(r\) in \(R\) if and only if \(T_n(R)\) is weakly normal for any positive integer \(n\), where \(T_n(R)\) is the \(n\times n\) upper triangular matrix ring over \(R\). And it follows that for a weakly normal ring \(R\), (1) \(R\) is Abelian if and only if \(R\) is strongly left idempotent reflexive; (2) \(R\) reduced if and only if \(R\) is \(n\)-regular; (3) \(R\) is strongly regular if and only if \(R\) is regular; (4) \(R\) is clean if and only if \(R\) is exchange; (5) exchange rings have stable range 1.