Modules whose endomorphism rings have isomorphic maximal left and right quotient rings

Let RM be a left R-module such that HomR(M, U) # 0 for any nonzero submodule U of M, let E(M) denote the injective hull of M, and let B (resp. A) denote the ring of R-endomorphisms of M (resp. E(M)). It is known that if M is nonsingular then B is left nonsingular and A is the maximal left quotient ring of B. We give here necessary and sufficient conditions on M for B to be right nonsingular and for A to be the maximal right quotient ring of B. 1. Ihtroduction and preliminaries. In [5], Utumi gave the solution of the following problem: given a ring S which is both left and right nonsingular, when is the maximal left quotient ring (MLQR) of S isomorphic to its maximal right quotient ring (MRQR)? He proved that this holds if and only if the converses of the nonsingular properties hold in S, namely, if and only if every left ideal of S, which has zero right annihilator, is essential in S and every right ideal of S, which has zero left annihilator, is essential in S [5, Theorem 3.3]. We consider here analogous questions for the endomorphism ring of an R-module. Let RM be a left R-module, where R is a ring with 1, and B = EndR(M) its ring of R-endomorphisms. The following notation will be used in the sequel: If U is a submodule of M, then IB(U) = {b E B: Mb C U}, rB(U) = {b E B: Ub =0}, IR(U) = {r E R: rU = 0}. If J is a right ideal of B, then lM(J) = {m E M: mJ = 0}. X C' Y means that X is an essential submodule of Y, i.e. X intersects nontrivially every nonzero submodule of Y; in case I is a left, right or two-sided ideal of a ring S, then jI C' ,S (resp. I, C' S,) will indicate that I is essential in S as a left (resp. right) ideal of S. Recalling that RM is said to be nonsingular in case the only submodule of M with essential (left) annihilator in R is the zero submodule-in our notation: IR(U) C' RR =? U = 0-we will call M cononsingular in case the only submodule of M with essential (right) annihilator in B is the zero submodule-i.e. rB(U) C' BB =? U = 0. Let E(M) denote the injective hull of M and A = EndR[E(M)] its ring of R-endomorphisms. It is known that if M is nonsingular then A is a (von Neumann) regular left, self-injective ring. If we impose a mild nondegeneracy Received by the editors June 23, 1981. 1980 Mathematics Subject Clasasfication. Primary 16A08, 16A65.