Two-sided essential submodules of Qr(R)

The focus of this paper are essential submodules, A, of the maximal right ring of quotients, Q(R)(r), of a right non-singular ring R. Since Q(r) is a R-R-bimodule, particular attention is given to submodules of Q(R)(r) which are also submodules of (R)Q(r). In this discussion, properties of R which are inherited by intermediate rings R subset of S subset of Q(r) are investigated. The results obtained are used to discuss homological properties of essential submodules A of Q(R)(r). In particular, the paper addresses the question when S-closed submodules of finite direct sums of copies of A are direct summands.