Prime ideals and finiteness conditions for Gabriel topologies over commutative rings (Q1201195)

The authors study the Gabriel topologies on rings with restriction either on ideals from that topologies or on chains of ideals from them. If \(P\) is a set of prime ideals of \(R\), then by \({\mathfrak C}(P)\) is denoted the Gabriel topology on \(R\) consisting from those ideals which are not contained in \(p\), where \(p\in P\). The Gabriel topologies of the form \({\mathfrak C}(P)\) where \(P\subseteq\text{Spec}(R)\) are called primal. A Gabriel topology is called of finite type (principal) if it has a base consisting from finitely generated (principal) ideals. It is proved (theorem 1.1) that a Gabriel topology \({\mathfrak C}\) is primal \(\Leftrightarrow{\mathfrak C}\) is an intersection of topologies of finite type \(\Leftrightarrow{\mathfrak C}\) is an intersection of principal topologies. A Gabriel topology \({\mathfrak C}\) is of finite type if and only if it satisfies the following condition: If the union of a totally ordered chain of ideals is in \({\mathfrak C}\), then one of the ideals must be in \({\mathfrak C}\) (theorem 1.2). The paper contains in terms of the Gabriel topologies a generalization of a theorem of \textit{I. S. Cohen} which says that if every prime ideal of a commutative ring \(R\) is finitely generated, then \(R\) is noetherian. It is proved, for example, that if every prime ideal from a Gabriel topology \({\mathfrak C}\) is finitely generated, then every ideal from \({\mathfrak C}\) is finitely generated (corollary 1). Special kinds of primal topologies are studied in the paper and some constructions of primal topologies are given. Some examples are given which show that a number of classes of Gabriel topologies are different.