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Summary: SIMPLICITY OF NONCOMMUTATIVE DEDEKIND DOMAINS
K. R. Goodearl and J. T. Stafford
Abstract. The following dichotomy is established: A nitely generated, complex
Dedekind domain that is not commutative is a simple ring. Weaker versions of this
dichotomy are proved for Dedekind prime rings and hereditary noetherian prime
rings.
Introduction
When the classical concept of a Dedekind domain was extended to noncommu-
tative rings, the natural examples that arose were either classical orders (and hence
nitely generated modules over their centres) or simple rings such as the Weyl alge-
bra A 1 (C ). Indeed, among nitely generated Dedekind domains over algebraically
closed elds, classical orders and simple rings are the only known examples. This
dichotomy in the examples suggests that an actual dichotomy might exist among
general Dedekind domains, although we are not aware that any such conjecture has
been formulated in the literature. The main goal in this paper is to establish just
such a result as well as give similar dichotomies for Dedekind prime rings and HNP
rings.
Before stating the rst theorem we need some de nitions. An HNP ring is simply
a (nonartinian) hereditary noetherian prime ring, while a Dedekind prime ring is
an HNP ring for which each nonzero ideal I is invertible in the sense that there
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