On uniform dimensions of ideals in right nonsingular rings Original Research Article

For any (S, R)-bimodule M, one can define an invariant d(M) by taking the supremum of n for which there exists a direct sum of nonzero subbimodules N = M1 circled plus M2 circled plus … circled plus Mn such that N is essential in M as a right 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 R terms of the ideal structure of Qmaxr(R) by working with the invariant d(I) = d(RIR) for ideals I subset of R. The family F(R) of ideals I for which there exists an ideal J subset of R with I circled plus J subset ofe Rr is characterized in various ways, and for I set membership, variant F(R), the invariant d(I) is related to the direct product decomposition of the ring E(IR) (injective hull) in Qmaxr(R). It is shown that d(I) is very well-behaved for the ideals I set membership, variant F(R) and various results are obtained on the relationship between d(I), u. dim(RIR) and u. dim(IR).