A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems


Let R be a ring and let C be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let V(C) denote a set of representatives of isomorphism classes in C and, for any module M in C, let [M] denote the unique element in V(C) isomorphic to M. Then V(C) is a reduced commutative semigroup with operation defined by [M] + [N] = [M ⊕ N], and this semigroup carries all information about direct-sum decompositions of modules in C. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if EndR(M) is semilocal for all M C, then V(C) is a Krull monoid. Suppose that the monoid V(C) is Krull with a finitely generated class group (for example, when C is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case, we study the arithmetic of V(C) using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid V(C) for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

Publication Title

Journal of Algebra and its Applications