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Lattice modules having small cofinite irreducibles. (English) Zbl 1013.06017
The authors continue the study of lattice modules for the following particular class. Let $$L$$ be a local Noether lattice with “maximal” element $$m$$, let $${\mathcal M}$$ be a Noetherian $$L$$-module with greatest element $$M$$. Then $${\mathcal M}$$ is said to have small cofinite irreducibles if for every positive integer $$n$$ there exists a meet-irreducible $$Q\in{\mathcal M}$$ such that $$Q\leq m^n M$$ and $${\mathcal M}/Q$$ is finite-dimensional. Several characterizations for such $$L$$-modules $${\mathcal M}$$ are given by means of $$m$$-primary elements of $${\mathcal M}$$ or the $$m$$-adic topology on $${\mathcal M}$$. It is also shown that $${\mathcal M}$$ has small cofinite irreducibles if and only if the $$L^*$$-module $${\mathcal M}^*$$ has cofinite irreducibles, where $$L^*$$ and $${\mathcal M}^*$$ were defined by the first two authors in Can. J. Math. 22, 327-331 (1970; Zbl 0197.29004). These results are applied to local Noetherian rings $$R$$ with maximal ideal $$m$$ and Noetherian $$R$$-modules $$M$$, the lattice of ideals of $$R$$ and the lattice of all $$R$$-submodules of $$M$$.
Reviewer: H.Mitsch (Wien)
##### MSC:
 06F10 Noether lattices
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