On uniform dimensions of ideals in right nonsingular rings

Abstract

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 = M 1 ⊕ M 2 ⊕ … ⊕ M n 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 Q max r ( R) by working with the invariant d( I) = d( R I R ) for ideals I ⊂ R. The family F( R) of ideals I for which there exists an ideal J ⊂ R with I ⊕ J ⊂ e R r is characterized in various ways, and for I ∈ F( R), the invariant d( I) is related to the direct product decomposition of the ring E( I R ) (injective hull) in Q max r ( R). It is shown that d( I) is very well-behaved for the ideals I ∈ F( R) and various results are obtained on the relationship between d( I), u. dim( R I R ) and u. dim( I R ).