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\(w\)-modules over commutative rings. (English) Zbl 1206.13005

The main purpose of this well written paper, is to extend the notion of \(w\)-modules to commutative rings with zero divisors. For this reason the authors introduced and studied the concepts of GV-ideals and GV-torsionfree modules. The methods employed in obtaining some results come from homological algebra, which are different from the methods used in the domain case. Let \(R\) be a commutative ring. An ideal \(J\) of a commutative ring \(R\) is called a Glaz-Vasconcelos ideal or a GV-ideal, denoted by \(J\in \mathrm{GV}(R)\), if \(J\) is finitely generated and the natural homomorphism \(\varphi:R\rightarrow\text{Hom}_R(J,R)\) is an isomorphism. An R-module \(M\) is called a GV-torsionfree module if whenever \(Jx=0\) for some \(J\in \mathrm{GV}(R)\) and \(x\in M\), then \(x=0\). Let \(M\) be a GV-torsionfree \(R\)-module. Then \(M\) is said to be a \(w\)-module if \(\text{Ext}^1_R(R/J,M)=0\) for any \(J\in \mathrm{GV}(R)\), and the \(w\)-envelope of \(M\) is defined by \(M_w=\{x\in E(M)| Jx\subseteq M\) for some \(J\in \mathrm{GV}(R)\}\), where \(E(M)\) is the injective envelope of \(M\).
As applications of the theory introduced in this paper is to give some new characterizations of \(w\)-Noetherian rings and Krull rings.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13D99 Homological methods in commutative ring theory
13E99 Chain conditions, finiteness conditions in commutative ring theory
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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