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

Abstract

Abstract. 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 ] , 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 End R (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.

Monoids of modules and arithmetic of direct-sum decompositions

“…See [5] for the proof of (1). (2) The third class of Krull monoids studied in this subsection are Krull monoids with finite cyclic class group having prime divisors in each class.…”

“…(b) If G P has a simple geometric structure (e.g., the set of vertices in a cube; see Examples 4.21 and 4.22), derive precise formulas for the arithmetical invariants, starting with the Davenport constant. A first result in this direction can be found in [5]. (c) Determine the extent to which the arithmetic of a Krull monoid with G P as in (b) is characteristic for G P .…”

The Davenport constant of a box

“…where 0 d is the origin in R d , and they prove a result that is reminiscent of our Theorem 5 (iii), see [3,Theorem 3.13]. Loosely speaking, they obtain the bounds…”

“…If n ≥ 3, we may additionally apply Lemma 4. Since, by assumption, x σ(2) = x σ(3) , we obtain that for i ∈ 1, n , the partial sums i l=1 x σ(l) are pairwise distinct and different from x σ(1) + x σ (3) . We obtain…”

“…and this is characteristic of cyclic groups, as for instance follows immediately from (3). For groups of rank at least four, equality is definitely not the rule, see [1,8,15].…”