Invertibility and Class Number of Orders

Abstract

This paper continues the work begun in [5] and concerns the invertibility of modules in a finite-dimensional, symmetric algebra L with 1 over a field. In particular, we continue the work done in [5] which dealt with the connection between invertibility in these algebras and a condition called the Brandt Condition, which is a reformulation by Kaplansky [6] of some ideas of Brandt.We begin by proving some preliminary results on invertibility and some equivalent conditions for the dual of an order to be principal.Then, we define the class number of an order and reformulate the concept of invertibility in terms of class number. In this terminology, we find some equivalent conditions which ensure that an order in certain algebras L (including commutative, symmetric algebras, and algebras with a strong involution) has class number equal to 1 (i.e., all modules principal), and we characterize a class of Brandt algebras over the quotient fields of valuation rings as those in which all orders have class number less than or equal to 2.