Pseudo valuation rings (Q2764640)

The author proves the analogue of some well known results regarding pseudo valuation domains for pseudo valuation rings. Let \(A\) be a commutative ring with identity. It is proved that if \(P \subseteq Q\) are non-zero prime ideals of \(A\) and if \(Q\) is strongly prime, then so is \(P.\) Using this, it is shown that \(A\) is a pseudo valuation ring, if and only if \(A\) is quasilocal with the maximal ideal \(M\) strongly prime. When this holds, then every overring \(B\) of \(A\) contained in the integral closure of \(A\) in its total quotient ring is a pseudo valuation ring with maximal ideal \(M.\)