Some prime ideals in the extensions of Noetherian rings (Q1261743)

Let \(R\) and \(S\) be commutative rings with identity and let \(f:R \to S\) be a homomorphism. An ideal \(Q\) of \(R\) is called \(S\)-prime \((S\)-primary) if for \(a\in R\) and \(s \in S\) with \(f(a)s \in QS\) either \(a \in Q\) or \(s \in QS\) \((s \in \sqrt {QS})\). \textit{D. L. McQuillan} [Arch. Math. 33, 121-126 (1979; Zbl 0456.13001)] introduced these notions and showed that if \(f\) is a flat homomorphism, then every prime ideal of \(R\) is \(S\)-prime, and that if \(R\) is an integrally closed domain and \(S\) is integral and torsion-free over \(R\), then every prime ideal of \(R\) is \(S\)-primary. Note that if \(QS=S\), then \(Q\) is \(S\)-prime. The author shows that for an ideal \(Q\) of \(R\) with \(QS \neq S\), \(Q\) is \(S\)-prime if and only if \(Q\) is a prime ideal of \(R\) and \(QS \cap R=Q\). Also, if \(QS \neq S\), \(Q\) is \(S\)- primary, then \(QS \cap R=Q\) and \(Q\) is a primary ideal of \(R\). The author defines an ideal \(Q\) to be \(S\)-quasi-primary if \(f(a)s \in QS\) with \(a \in R\) and \(s \in S\) implies \(a \in \sqrt Q\) or \(s \in QS\). Suppose that \(QS \neq S\). It is shown that \(Q\) is \(S\)-quasi-primary if and only if \(\widetilde Q=QS \cap R\) is a primary ideal of \(R\) such that \(\sqrt Q=\sqrt {\widetilde Q}=P \in \text{Spec} (R)\) and \(\text{Ass}_ R (S/QS)=\{P\}\). A number of other related results are given, especially in the case where \(R\) and \(S\) are Noetherian.