The quotient between length and multiplicity : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University

Abstract

This dissertation examines the finiteness of the algebraic invariants nA(M) and θA(M). These invariants, based on the ratio of length and multiplicity and the ratio of Loewy length and multiplicity respectively, are studied in general and under certain conditions. The finiteness of θA(M) is established for a large class of algebraic structures. nA(M) is shown to be finite in the low dimensional case as well as when we restrict our attention to special sets of ideals. Also considered in this dissertation are equivalent conditions for the local case to be bounded by the graded case when evaluating nA(M).