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Eakin-Nagata-Eisenbud theorem for right \(S\)-Noetherian rings. (English) Zbl 1529.16018

All rings considered in this paper under review are associative with unity and all modules are unitary right modules. The authors consider an \(S\)-version of Eakin-Nagata-Eisenbud theorem for general rings (not necessarily commutative).
Recall that a submodule \(N\) of a right \(R\)-module \(M\) is called \(S\)-finite if \(Ns\subseteq F\subseteq N\) for an element \(s\in S\) and a finitely generated submodule \(F\) of \(M,\) where \(S\) is a multiplicative subset of \(R.\) M is \(S\)-Noetherian if every submodule of \(M\) is \(S\)-finite. A ring \(R\) is called right \(S\)-Noetherian if \(R_{R}\) is \(S\)-Noetherian.
The authors show first that if \(E\) is a ring extension of \(R,\) \(M\) a right \(E \)-module and \(S\) a multiplicative subset of \(R,\) and if \(M\) is \(S\)-Noetherian as a right \(R\)-module then \(M\) is \(S\)-Noetherian as a right \(E\)-module. In particular, if \(R\) is right \(S\)-Noetherian and \(M\) is \(S\)-finite as a right \(R\)-module then \(M\) is \(S\)-Noetherian as a right \(E\)-module.
Next they prove that if \(E\) is an \(S\)-finite normalizing ring extension of \( R,\) where \(S\) is a multiplicative subset of \(R\) (that is there exist \(s\in S\) and \(e_{i}\in E\) such that \(Es\subseteq e_{1}R+\cdots+e_{k}R\subseteq E\) with \( e_{1}=1_{R}=1_{E}\) and \(e_{i}R=Re_{i}\) for all \(i=1,\ldots,k)\) and if \(E\) is a right \(S\)-Noetherian ring and \(M\) is \(S\)-finite as a right \(E\)-module then \( M \) is \(S\)-Noetherian as a right \(R\)-module. In particular if \(E\) is a right \(S \)-Noetherian ring then \(R\) is a right \(S\)-Noetherian ring.
An \(S\)-version of Cohen theorem is also given. Moreover, they show that every right \(S\)-Noetherian domain is a right Ore domain.
Applications to composite polynomial, composite power series and composite skew polynomial rings are given in order to be right \(S\)-Noetherian.

MSC:

16P99 Chain conditions, growth conditions, and other forms of finiteness for associative rings and algebras
16S20 Centralizing and normalizing extensions
16D25 Ideals in associative algebras
16U20 Ore rings, multiplicative sets, Ore localization
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References:

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