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Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids

✍ Scribed by Alberto Facchini

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 264 KB
Volume: 256
Category: Article
ISSN: 0021-8693
DOI: 10.1016/s0021-8693(02)00164-3

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✦ Synopsis

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 (C) of isomorphism classes of a class 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 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 (C) as a subdirect product of free commutative monoids.

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