| This dissertation is concerned with several problems in ring theory that fall under the heading of representation theory of associative algebras. The first two of these are in matrix theory; the third is in integral representation theory.;For the first, let F be a field, and let M n(F) be the ring of n by n matrices over F. If S is a finite set of matrices in Mn(F), let l(S) denote the smallest integer i such that FSi = Mn( F). (The notation Si denotes the set of all words in S of length at most i.) The problem is to find a bound for l(S) that depends only on n. In Chapter 2, we prove that lS∈On n . Other results of Chapter 2 include improved bounds for l (S) when S contains matrices of a special type and an application to affine semiprime algebras of GK dimension 1.;In Chapter 3, we consider the following problem: When can we find matrices A1,&ldots;,Am∈Mn F whose pencil x1A1+&cdots;+xmAm ∈MnFx1 ,&ldots;,xm has prescribed minimal polynomial &phis;(z)? We construct an associative F-algebra C&phis; whose representation theory is closely linked to the existence of such matrices. The main result of Chapter 3 says that, when &phis;(z) is quadratic, we can find matrices as above if and only if n is an appropriate power of 2.;In Chapter 4, we study the genus class groups of modules over certain one-dimensional rings. Specifically, let k be a field and let Lambda be an affine, semiprime noetherian k-algebra of pi degree d and Krull dimension 1. We prove that if for some maximal order Gamma containing Lambda there exists a conductor ideal C which is semiprime, then there exists a positive integer m such that Cl(Lambda)n is finite for all n prime to m. Other results of Chapter 4 include similar statements for genus class groups of lattices over such Lambda, and a removal of the hypothesis on C when k has positive characteristic or is a number field. |
