## Deep decompositions of modules.(English)Zbl 0937.13003

Let $$R$$ be a commutative ring with identity. The author defines a decomposition of an $$R$$-module into submodules, $$M=\bigoplus M_\alpha$$, to be deep if, given any submodule $$H$$ of $$M$$, $$H=\bigoplus (H\cap M_\alpha)$$. To describe (part of) his two main results we need some notation. First let $$\text{M Spec }R$$ denote the maximal spectrum of $$R$$, let $$S(M)$$ denote the support of the module $$M$$, let $$V(I)=\{P\in \text{Spec} R |I\subseteq P\}$$ for any ideal $$I$$ of $$R$$, for any subset $$W$$ of $$\text{M Spec }R$$ let $$M^W= \{m\in M|V(\text{Ann} m) \cap\text{M Spec} R \subseteq W\}$$ and $$W_*= \text{M Spec} R \setminus W$$, let $$M^P=M^{\{P\}}$$ and $$M^{P_*}= M^{\{P\}_*}$$ for any $$P\in\text{M Spec} R$$, and let $$^AM=\{m\in M|V(\text{Ann} m) \subseteq A\}$$ for any subset $$A$$ of $$\text{Spec }R$$.
Theorem A shows: (i) If $$M=N\oplus K$$ is a decomposition of $$M$$ then this decomposition is deep iff $$S(N)\cap S(K)=\emptyset$$ iff for any nonzero $$m\in M$$, the ring $$R/(\text{Ann} m)$$ is a direct sum $$T_1\oplus T_2$$ such that if $$p\in S(N)\cap V(\text{Ann} m)$$ then $$p/(\text{Ann} m)$$ has the form $$p'\oplus T_2$$ for some $$p'\in \text{Spec} T_1$$ and likewise for $$p\in S(K)\cap V(\text{Ann} m)$$;
(ii) if the two subsets $$A$$ and $$B$$ partition $$S(M)$$ such that, for every $$m\in M$$, $$A\cap V(\text{Ann} m)$$ and $$B\cap V(\text{Ann} m)$$ are both closed subsets of $$\text{Spec} R$$, then $$^AM\oplus {^BM}$$ is a deep decomposition of $$M$$.
Part of theorem B shows that the following are equivalent: (a) $$M= \bigoplus M^P$$ over $$P\in\text{M Spec }R$$;
(b) for any nonzero $$m\in M$$, $$R/(\text{Ann} m)$$ is a direct sum of finitely many quasi-local rings;
(c) $$M=M^W \oplus M^{W_*}$$ for any (closed) subset $$W$$ of $$\text{M Spec }R$$;
(d) $$M=M^P \oplus M^{P_*}$$ for any $$P\in\text{M Spec }R$$.
Moreover, $$\bigoplus M^P$$ over $$P\in\text{M Spec }R$$ is always a deep decomposition.
Reviewer: J.Clark (Dunedin)

### MSC:

 13C05 Structure, classification theorems for modules and ideals in commutative rings 13A15 Ideals and multiplicative ideal theory in commutative rings
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### References:

  Fuchs L., Glasgow-Math. J 38 pp 321– (1996) · Zbl 0896.13012  Maths E., Memoirs Amer. Math. Soc 49 (1964)
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