Characterization theorems of \(S\)-Artinian modules. (English) Zbl 1488.13060
Let \(A\) be a commutative (associative) ring with a unit \(1 \neq 0\), \(S\) a multiplicatively closed subset (abbr., m.c.s.) of \(A\), and let \(X\) be a nonzero unital \(A\)-module. For any submodule \(Y\) of \(X\), the residual \((Y\colon _AX)\) of \(Y\) by \(X\) is defined to be the ideal \(\{r \in A\colon rX \subseteq Y\}\); the annihilator ann\((X)\) of \(X\) is defined to be the ideal \((0\colon _A X)\). We say that \(X\) is a multiplication module if every submodule \(Y _1\) of \(X\) has the form \(Y = (Y\colon _A X)X\) (see [Z. A. El-Bast and P. F. Smith, Commun. Algebra 16, No. 4, 775–779 (1988; Zbl 0642.13003); P. F. Smith, Arch. Math. 50, No. 3, 223–235 (1988; Zbl 0615.13003)].
The paper under review deals with the study of \(S\)-Artinian modules and finitely \(S\)-cogenerated modules. The module \(X\) is said to be \(S\)-Artinian module if in each descending sequence \(\Lambda _n\), \(n \in \mathbb{N}\), of submodules of \(X\), there exist \(s \in S\) and \(k \in \mathbb{N}\), such that \(s\Lambda _k \subseteq \Lambda _n\), \(n \ge k\); \(X\) is called finitely S-cogenerated if for each family \(\{Y _i\} _{i \in \Delta}\) of submodules of \(X\), \(\cap _{i \in \Delta} Y _i = \{0\}\) implies \(s(\cap _{i \in \Delta '} Y _i) = \{0\}\), for some \(s \in S\) and a finite subset \(\Delta ^{\prime} \subseteq \Delta \). In these terms, the usual notions of an Artinian module and of a finitely cogenerated module correspond to the special case where \(s = 1\). The reviewed paper gives many examples, properties and \(S\)-versions of several known results. It provides several characterizations of \(S\)-Artinian modules. One of these states that \(X\) is \(S\)-Artinian if and only if every factor \(A\)-module \(X/Y\) is finitely \(S\)-cogenerated. Also, the paper shows that if \(X\) is a multiplication module and \(\mathrm{ann}_A(X)\) is a prime ideal of \(A\), such that \(\mathrm{ann}_A(X) \cap S = \emptyset \), then \(X\) is a finitely \(S\)-cogenerated module if and only if its zero submodule is strongly prime. Finally, it proves that the following conditions are equivalent: (i) \(X\) is finitely cogenerated; (ii) \(X\) is finitely \((A \setminus P)\)-cogenerated, for each prime ideal \(P\) of \(A\); (iii) \(X\) is \((A \setminus M)\)-cogenerated whenever \(M\) is a maximal ideal of \(A\).
The paper under review deals with the study of \(S\)-Artinian modules and finitely \(S\)-cogenerated modules. The module \(X\) is said to be \(S\)-Artinian module if in each descending sequence \(\Lambda _n\), \(n \in \mathbb{N}\), of submodules of \(X\), there exist \(s \in S\) and \(k \in \mathbb{N}\), such that \(s\Lambda _k \subseteq \Lambda _n\), \(n \ge k\); \(X\) is called finitely S-cogenerated if for each family \(\{Y _i\} _{i \in \Delta}\) of submodules of \(X\), \(\cap _{i \in \Delta} Y _i = \{0\}\) implies \(s(\cap _{i \in \Delta '} Y _i) = \{0\}\), for some \(s \in S\) and a finite subset \(\Delta ^{\prime} \subseteq \Delta \). In these terms, the usual notions of an Artinian module and of a finitely cogenerated module correspond to the special case where \(s = 1\). The reviewed paper gives many examples, properties and \(S\)-versions of several known results. It provides several characterizations of \(S\)-Artinian modules. One of these states that \(X\) is \(S\)-Artinian if and only if every factor \(A\)-module \(X/Y\) is finitely \(S\)-cogenerated. Also, the paper shows that if \(X\) is a multiplication module and \(\mathrm{ann}_A(X)\) is a prime ideal of \(A\), such that \(\mathrm{ann}_A(X) \cap S = \emptyset \), then \(X\) is a finitely \(S\)-cogenerated module if and only if its zero submodule is strongly prime. Finally, it proves that the following conditions are equivalent: (i) \(X\) is finitely cogenerated; (ii) \(X\) is finitely \((A \setminus P)\)-cogenerated, for each prime ideal \(P\) of \(A\); (iii) \(X\) is \((A \setminus M)\)-cogenerated whenever \(M\) is a maximal ideal of \(A\).
Reviewer: Ivan D. Chipchakov (Sofia)
MSC:
| 13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |
| 16P20 | Artinian rings and modules (associative rings and algebras) |
| 13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |
