Reflexive primes, localization and primary decomposition in maximal orders

Abstract

If R is a maximal order and P a reflexive prime ideal of R, then the Goldie localization of R at P is shown to be the (partial) quotient ring of R with respect to the Ore set C(P) = {r E RIrx E P X x E P). This is accomplished by introducing new symbolic powers of the prime P which agree with Goldie's symbolic powers. As a consequence, whenever P is a reflexive prime ideal of R and p(n) the nth (Goldie) symbolic power of P, then an ideal B is reflexive if and only if B= n,= Pl(n') for uniquely determined reflexive primes Pi and integers ni > 0. More generally, each bounded essential right (left) ideal is shown to have a reduced primary decomposition and an explicit determination of the components is given in terms of the bound of the ideal. 1. In [67], Goldie gave a method for constructing a local ring Rp given an arbitrary two-sided Noetherian ring R and a prime ideal P, which for commutative R coincides with the usual commutative localization at P. Just as in the commutative case, Goldie's localization at P utilizes the set C(P) = {x E RI x + P is regular in R/P}. There are many instances where C(P) is an Ore set: R any commutative Noetherian ring, R an Asano order (Michler [69], Hajarnavis and Lenagan [71], Kuzmanovich [72]), R a hereditary Noetherian prime ring with P nonidempotent (Chatters and Ginn [72]), R the enveloping algebra of finite dimensional nilpotent Lie algebra (McConnell [68]) and R = AG, A a Noetherian prime ring of characteristic 0, G a finite group and P the augmentation ideal of R (Michler [72]). The object of the first section of this paper is to show that whenever R is a Noetherian maximal order and P a reflexive prime, C(P) is an Ore set of regular elements of R. Such rings include Asano orders and most classical maximal orders. To accomplish this, we first define a new symbolic power, denoted p(n), for a reflexive prime ideal of R. Actually, all of the results of this paper are valid for nonmaximal R and reflexive primes P satisfying P* P 0 P and PP* C R. Next, we establish the basic properties of these symbolic powers, ultimately showing that they are identical to the Hn of Goldie [67] (see also Michler [72] for another equivalent formulation). Although we use the Received by the editors February 19, 1975 and, in revised form, October 20, 1975. AMS (MOS) subject classifications (1970). Primary 16A08, 16A66; Secondary 16A 10, 16A46.