Ideal arithmetic in noetherian Pi rings

Abstract

A domain satisfying the above equivalent characterisations is called a Dedekind domain. In the setting of non-commutative orders, the corresponding theory was axiomatised by Asano [2, 31 and further relined by Michler [t l] and Robson 1131. The starting point of this theory is condition (b) and indeed, in modern terms the rings considered by Asano can be thought of as prime Noetherian rings satisfying a polynomial identity and condition fb). However, while condition (a) is often taken to be the definition of a commutative Dedekind domain [ 10, p. 294; 15, p. 2701, its implications in non-commutative rings have not so far been studied. It is the aim of this paper to do so. One obstruction to following one of the standard proofs of the commutative theory is that if P, P, 1.. P, is an invertible product of prime ideals then it is not immediately clear that each Pi is individually 475 0021-8693/89 $3.00