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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:

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