Factorization Theory and Decompositions of Modules

Abstract

Let R be a commutative ring with identity. It often happens that M₁ ⊕ ⋯ ⊕ M_s ≅ N₁ ⊕ ⋯ ⊕ N_t for indecomposable R-modules M₁, …, M_s and N₁, …, N_t 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]. 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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Notes on contributors

Nicholas R. Baeth

ROGER WIEGAND received his Ph.D. from the University of Washington in 1967. He has been at the University of Nebraska since 1972, and, as of 2011, is Willa Cather Professor Emeritus. He retired from teaching in order to have more time for research, travel, and climbing. He has had visiting positions at Connecticut, Wisconsin, Purdue, and MSRI (Berkeley). Sixteen students have completed Ph.D.s under his direction, and he has two more in the works.

Roger Wiegand

NICHOLAS R. BAETH received a B.S. in mathematics and a B.A. in computer science in 2000 from Pacific Lutheran University. He earned his Ph.D. in mathematics from the University of Nebraska–Lincoln in 2005, under the direction of Roger Wiegand. Since then he has been a faculty member at the University of Central Missouri where he is now an associate professor of mathematics. In 2013 he will serve as the NAWI-Graz Fulbright Visiting Professor in the Natural Sciences and will teach a course on the interconnections of factorization theory and direct-sum decompositions.