## 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)].

### MSC:

 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

### Citations:

Zbl 0573.13001; Zbl 0579.13002; Zbl 0601.13004; Zbl 0183.040
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### References:

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