Monoids of modules and arithmetic of direct-sum decompositions

Nicholas R. Baeth and Alfred Geroldinger

Vol. 271 (2014), No. 2, 257–319

DOI: 10.2140/pjm.2014.271.257

Abstract

Let $R$ be a (possibly noncommutative) ring and let $\mathcal{C}$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal{V}\left(\mathcal{C}\right)$ of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in $\mathcal{C}$. If the endomorphism ring of each module in $\mathcal{C}$ is semilocal, then $\mathcal{V}\left(\mathcal{C}\right)$ is a Krull monoid. Although this fact was observed nearly a decade ago, the focus of study thus far has been on ring- and module-theoretic conditions enforcing that $\mathcal{V}\left(\mathcal{C}\right)$ is Krull. If $\mathcal{V}\left(\mathcal{C}\right)$ is Krull, its arithmetic depends only on the class group of $\mathcal{V}\left(\mathcal{C}\right)$ and the set of classes containing prime divisors. In this paper we provide the first systematic treatment to study the direct-sum decompositions of modules using methods from factorization theory of Krull monoids. We do this when $\mathcal{C}$ is the class of finitely generated torsion-free modules over certain one- and two-dimensional commutative Noetherian local rings.

Keywords

Krull monoids, sets of lengths, direct-sum decompositions, indecomposable modules

Mathematical Subject Classification 2010

Primary: 13C14, 16D70, 20M13

Milestones

Received: 27 June 2013

Accepted: 16 December 2013

Published: 20 September 2014