A generalization of strongly regular rings (Q790909)

A ring \(A\) is s-weakly regular if for all \(a\) in \(A\) \(a\) is in \(aAa^ 2A\). The class of s-weakly regular rings lies strictly between the class of strongly regular rings and the class of weakly regular rings. Just as strongly regular rings are the reduced regular rings, the s-weakly regular rings are the reduced weakly regular rings. A ring \(A\) is s-weakly regular if and only if \(A\) is reduced and every proper prime ideal is maximal. In this case every proper prime ideal must be completely prime. It is shown that there is a maximal s-weakly regular ideal \(S(A)\) in every ring. \(S(A)\) is shown to be a hereditary radical.