Factorization Theory and Decompositions of Modules

Author: Baeth Nicholas R.   Wiegand Roger  

Publisher: Mathematical Association of America

ISSN: 1930-0972

Source: American Mathematical Monthly, Vol.120, Iss.1, 2013-01, pp. : 3-34

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Abstract

Let R be a commutative ring with identity. It often happens that M1 ⊕ · · · ⊕ Ms ≅ = N1 ⊕ · · · ⊕ Nt for indecomposable R-modules M1, . . . , Ms and N1, . . . , Nt with st. This behavior can be captured by studying the commutative monoid {[M] | M is an R -module } of isomorphism classes of R-modules with operation given by [M] + [N] = [MN]. In this mostly self-contained exposition, we introduce the reader to the interplay between the the study of direct-sum decompositions of modules and the study of factorizations in integral domains.