Monoids, direct-sum decompositions, and elasticity of factorizations.

摘要

Research into the factorization properties of monoids has its roots in the study of the multiplicative properties of integral domains. A fundamental problem is to measure the extent to which unique factorization into irreducible elements can fail for a given integral domain. Not only can an element have several distinct factorizations, but even the number of factors can vary. One measures this variability by means of the elasticity function.; Factorization problems arise also in representation theory. Given a finitely generated module M over a local ring R, let +(M) denote the monoid (with ⊕ as the operation) of isomorphism classes of finitely generated modules that are direct summands of direct sums of copies of M. For the case of a one-dimensional local ring R, it is shown in Chapter 3 of this dissertation that +(M) is isomorphic to a Diophantine monoid, that is, to the monoid of nonnegative integer solutions to a suitable homogeneous system of linear equations with integer coefficients. It is known, conversely, that every Diophantine monoid actually arises in this fashion. The question then arises as to whether there exists a single one-dimensional local ring R whose finitely generated modules can represent every Diophantine monoid. In Chapter 3 we give a negative answer to this question.; In Chapter 4 we use the theory of Krull monoids, divisor class groups, and block monoids to obtain new results on the elasticity of Diophantine monoids. We show, for example, that if the divisor class group of a Diophantine monoid is cyclic of prime power order, then the most extreme failure of unique factorization occurs in a particularly simple way. Chapter 5 contains a description of how we compute the elasticity and investigates a new way, hinted at in the literature, of finding the elasticity of a Diophantine monoid without computing Gröbner bases.