William H. Gustafson

Roth's theorems over commutative rings

Abstract In 1952, W.E. Roth showed that matrix equations of the forms AX − YB = C and AX − XB = C over fields can be solved if and only if certain block matrices built from A, B , and C are equivalent or similar. We show here that these criteria remain valid over arbitrary commutative rings. To do this, we use standard commutative algebra methods to reduce to the case of Artinian rings, where a simple argument with

A note on matrix inversion

Abstract We generalize a recent result of Thompson on inverses of block matrices over principal ideal domains to isomorphisms of direct sums of modules over an arbitrary ring.

On matrix equivalence and matrix equations

Abstract Roths theorem on the solvability of matrix equations of the form AX−YB=C is proved for matrices with coefficients from a division ring or a ring which is module-finite over its center.

Artin rings of global dimension two

In this paper, we show that a left artin ring A of global dimension at most two has Loewy length at most 2” - 1, where n is the number of simple components of A/rad(A). We assume that the reader is familiar with the trends (but not necessarily the details) of contemporary representation theory of artin rings. Let us record a few conventions. The rings discussed here will be left artinian, unless otherwise specified, and will sometimes be finite-dimensional algebras over an algebraically closed field, henceforth denoted by k. Modules will be finitely generated left modules unless otherwise specified. The radical of A will usually be denoted by r; the Loewy

Matrix Completions Over Dedekind Rings

We show that if M is a k × n matrix over a Dedekind domain R, with k < n. and if d is an element of the ideal of R generated by the k × k minors of M, then M forms the top k rows of an n × n matrix of determinant d. This continues a program initiated by Hermite in 1849.

Reflexive serial algebras

Abstract An algebra A of n × n matrices over a field K is reflexive if every element of M n ( K ) that leaves invariant all invariant subspaces of A is in A . We characterize those basic, indecomposable serial algebras that are reflexive.

Quivers and matrix equations

Abstract We discuss methods for solving some familiar matrix equations. The methods were derived from the theory of quivers, a powerful tool in the representation theory of algebras. It is not necessary that the reader understand quivers, but we include a brief explanation for those who may be interested.

Modules and matrices

Abstract Noncommutative ring theory was described in terms of matrices in its earliest days; we give some examples showing how matrix theory can now in turn profit from the theory of rings and modules.

A question of Guralnick about reflexive algebras

Abstract Let A be a commutative, finite-dimensional, local k -algebra, and set B = End A ( M ), where M is a finitely generated, faithful, indecomposable A -module. Guralnick asked whether B must equal A if B is an intermediate algebra of A , i.e., if the A - and B -submodules of M are the same. We give a class of examples where this is not the case.