Fitting ideals and module structure

Abstract

Let R be a commutative ring with a 1. Original work by H. Fitting showed how we can associate to each finitely generated E-module a unique sequence of R-ideals, which are known as Fitting Ideals. The aim of this thesis is to undertake an investigation of Fitting Ideals and their relation with module structure and to construct a notion of Fitting Invariant for certain non-commutative rings. We first of all consider the commutative case and see how Fitting Ideals arise by considering determinantal ideals of presentation matrices of the underlying module and we describe some applications. We then study the behaviour of Fitting Ideals for certain module structures and investigate how useful Fitting Ideals are in determining the underlying module. The main part of this work considers the non-commutative case and constructs Fitting Invariants for modules over hereditary orders and shows how, by considering maximal orders and projectives in the hereditary order, we can obtain some very useful invariants which ultimately determine the structure of torsion modules. We then consider what we can do in the non-hereditary case, in particular for twisted group rings. Here we construct invariants by adjusting presentation matrices which generalises the previous work done in the hereditary case.