MONOIDS OF MODULES AND ARITHMETIC OF DIRECT-SUM DECOMPOSITIONS

摘要

Let R be a (possibly noncommutative) ring and let C be a class of finitely generated (right)-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set V(C) 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 C. If the endomorphism ring of each module in C is semilocal,then V(C) is a Krull monoid. Although this fact was observed nearly a decade ago, the focus of study thus far has been on ringand module-theoretic conditions enforcing that V(C) is Krull. If V(C) is Krull, its arithmetic depends only on the class group of V(C) 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 C is the class of finitely generated torsion-free modules over certain one- and two-dimensional commutative Noetherian local rings.