When every regular ideal is \(S\)-finite. (English) Zbl 07842195
Let \(R\) be a commutative ring with nonzero identity and \(S\subseteq R\) be a multiplicatively closed subset. An ideal \(I\) of \(R\) is called \(S\)-finite if there exists a finitely generated subideal \(J\subseteq I\) and \(s\in S\) such that \(sI\subseteq J\). Then \(R\) is called an \(S\)-Noetherian ring if every ideal of \(R\) is \(S\)-finite.
In the paper under review the authors introduced and studied the notion of regular \(S\)-Noetherian rings that is rings such that every regular ideal is \(S\)-finite. This notion generalized regular-Noetherian rings (every regular ideal is finitely generated) and \(S\)-Noetherian rings. If \(R\) is an integral domain these two notion are identical. The property of regular \(S\)-Noetherian rings is investigated in particular Pullback constructions that is \(D+M\)-constructions and trivial extension and amalgamated duplication of a ring along an ideal. Several example are given.
In the paper under review the authors introduced and studied the notion of regular \(S\)-Noetherian rings that is rings such that every regular ideal is \(S\)-finite. This notion generalized regular-Noetherian rings (every regular ideal is finitely generated) and \(S\)-Noetherian rings. If \(R\) is an integral domain these two notion are identical. The property of regular \(S\)-Noetherian rings is investigated in particular Pullback constructions that is \(D+M\)-constructions and trivial extension and amalgamated duplication of a ring along an ideal. Several example are given.
Reviewer: Parviz Sahandi (Tabriz)
MSC:
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13B99 | Commutative ring extensions and related topics |
| 13E15 | Commutative rings and modules of finite generation or presentation; number of generators |
Keywords:
\(S\)-finite ideal; \(S\)-Noetherian ring; reg-\(S\)-Noetherian ring; trivial ring extension; amalgamation algebra along an idealReferences:
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