On open ring pairs of commutative rings. (English) Zbl 1101.13009
A commutative domain \(D\) is called an open domain if the canonical contraction map Spec\((E)\longrightarrow\) Spec\((D), \,P \mapsto P\cap D,\,\) is an open map relative to the Zariski topology for each overring \(E\) of \(D\). A commutative ring \(A\) is called an open ring if \(A/P\) is an open domain for each \(P\in \text{Spec} (A)\). An open ring pair is a pair \((R,T)\) of commutative rings with identity elements such that \(R\) is a unital subring of \(T\) and \(S\) is an open ring for each subring \(S\) of \(T\) such that \(R\subseteq S\subseteq T\).
The authors prove that if \(R\subseteq T\) is an integral extension of commutative unital rings such that \(R\) is an open ring, \(R[a,b]\) is a going-down ring for each \(a,\,b\in T\) and \(T\) is semiquasilocal, then \((R,T)\) is an open ring pair. It is also proved that if \(R\subseteq T\) is an extension of commutative unital rings such that \(R[a,b]\) is an open ring for each \(a,\,b\in T\), then \((R,T)\) is an INC-pair.
The authors prove that if \(R\subseteq T\) is an integral extension of commutative unital rings such that \(R\) is an open ring, \(R[a,b]\) is a going-down ring for each \(a,\,b\in T\) and \(T\) is semiquasilocal, then \((R,T)\) is an open ring pair. It is also proved that if \(R\subseteq T\) is an extension of commutative unital rings such that \(R[a,b]\) is an open ring for each \(a,\,b\in T\), then \((R,T)\) is an INC-pair.
Reviewer: Toma Albu (Istanbul)
MSC:
13B24 | Going up; going down; going between (MSC2000) |
13A15 | Ideals and multiplicative ideal theory in commutative rings |
14A05 | Relevant commutative algebra |
13G05 | Integral domains |
13B30 | Rings of fractions and localization for commutative rings |
13B21 | Integral dependence in commutative rings; going up, going down |
13E05 | Commutative Noetherian rings and modules |