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Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids. (English) Zbl 1016.16002

Author’s abstract: Commutative monoids yield an analogy between the theory of factorization in commutative integral domains and the theory of direct sum decompositions of modules. We show that the monoid \(V(\mathcal C)\) of isomorphism classes of a class \(\mathcal C\) of modules with semilocal endomorphism rings is a Krull monoid (Theorem 3.4). Krull monoids often appear in the study of factorizations of elements in integral domains, and are defined as the monoids \(V\) for which there is a divisor homomorphism of \(V\) into a free commutative monoid. In particular, we consider the case in which \(\mathcal C\) is the class of biuniform modules. For this class the validity of a weak form of the Krull-Schmidt Theorem is explained via a representation of \(V(\mathcal C)\) as a subdirect product of free commutative monoids.

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
13A05 Divisibility and factorizations in commutative rings
20M14 Commutative semigroups
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