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Factorization Theory and Decompositions of Modules

Nicholas R. Baeth and Roger Wiegand
The American Mathematical Monthly
Vol. 120, No. 1 (January 2013), pp. 3-34
DOI: 10.4169/amer.math.monthly.120.01.003
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Page Count: 32
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Factorization Theory and Decompositions of Modules
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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 s ≠ t. 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] = [M ⊕ N*#x5d;. 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.

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