Some remarks on multiplication modules. (English) Zbl 0615.13003

Let R be a commutative ring with identity and M a unital R-module. The module M is called a multiplication module provided for each submodule N of M there exists an ideal I of R such that \(N=IM\). Throughout the paper the following characterization of multiplication modules is exploited: M is a multiplication module if and only if for each maximal ideal P of R such that \(M_ P\neq 0\) there exist \(p\in P\) and \(m\in M\) with (1- p)M\(\subseteq Rm\). If M is a sum \(\sum _{\lambda \in \Lambda}M_{\lambda}\quad of\) its submodules \(M_{\lambda} (\lambda \in \Lambda)\) then M being a multiplication module can be characterized in various ways, for example M is a multiplication module if and only if \(M_{\lambda}=(M_{\lambda}:M)M\), \(\lambda\) \(\in \Lambda\), where \((M_{\lambda}:M)=\{r\in R: rM\subseteq M_{\lambda}\}\). In this case \(N=\sum _{\lambda \in \Lambda}(N\cap M_{\lambda})\quad for\) any submodule N of M. If \(N_ i\) (1\(\leq i\leq k)\) is a finite collection of submodules of an R-module M such that \(N_ i+N_ j\) is a multiplication module for all \(1\leq i<j\leq n\) then \(N_ 1+...+N_ k\) is a multiplication module, and, in addition, \(N_ 1,...,N_ k\) are all multiplication modules if and only if \(N_ 1\cap...\cap N_ k\) is a multiplication module. If M is a multiplication module with annihilator J and A, B ideals of R, then AM\(\subseteq BM\) if and only if \(A\subseteq B+J\) or \(M=(B+J):A)M.\)
It is known that any projective ideal is a multiplication module [see W. W. Smith, Can. J. Math. 21, 1057-1061 (1969; Zbl 0183.040)]. It is proved here that any finitely generated multiplication module whose annihilator is generated by an idempotent is projective. An estimate is given for the number of generators of (N:M), where N is a submodule of a multiplication module M, in terms of the numbers of generators of N, M and the annihilator of M. Some of the results of this paper generalize work of A. G. Naoum and M. A. K. Hasan [Arab J. Math. 4, 59- 75 (1983; Zbl 0601.13004) and Arch. Math. 46, 225-230 (1986; Zbl 0573.13001)].


13C10 Projective and free modules and ideals in commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13A05 Divisibility and factorizations in commutative rings
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