ON UNIFORM DIMENSIONS OF IDEALS IN RIGHT NONSINGULAR RINGS

For any (S,R)-bimodule M, one can define an invariant d(M) by taking t he supremum of n for which there exists a direct sum of nonzero subbim odules N = M-1 + M-2 + ... + M-n such that N is essential in M as a ri ght R-submodule. This invariant is a sort of hybrid between the right uniform dimension and the 2-sided uniform dimension. In this paper, we study the ideal structure of a right nonsingular ring Ii terms of the ideal structure of Q(max)(r)(R) by working with the invariant d(I) = d(I-R(R)) for ideals I subset of R. The family F(R) of ideals I for wh ich there exists an ideal J subset of R with 1 + J subset of(e) R-R is characterized in various ways, and for I is an element of F(R), the i nvariant d(I) is related to the direct product decomposition of the ri ng E(I-R) (injective hull) in Q(max)(r)(R). It is shown that d(I) is v ery well-behaved for the ideals I is an element of F(R) and various re sults are obtained on the relationship between d(I), u. dim(I-R(R)) an d u. dim(I-R). (C) 1998 Elsevier Science B.V. All rights reserved.