Quotient Rings of Endomorphism Rings of Modules with Zero Singular Submodule

Throughout this paper («, M, N, S) will denote a Morita context satisfying a certain nonsingularity condition.For such contexts we give necessary and sufficient conditions in terms of M and « for S to have a semisimple maximal left quotient ring; respectively a full linear maximal left quotient ring, a semisimple classical left quotient ring.In doing so we extend the corresponding well-known theorems for rings (employing them in the process) to endomorphism rings. Suppose (R, M, N, S) is a Morita context ([1], [2]).That is suppose RMS and SNR are bimodules with an R-R bimodule homomorphism ( , ) : M ®s N^R and an S-S bimodule homomorphism [, ] : N ®R M->S satisfying mAjty, m2] = (/»!, nf)m2 and nY(mx, n2) = [nlt mx]n2 for all mlt m2e N and n,, n2 e N.Throughout, unless otherwise indicated, M and JV will satisfy the following condition: Ms is faithful; and [N,m]=0 for me M implies that m=0.Note that when this condition is satisfied, we can (and will) assume that SçHomR(M, M).Let RM be any left /(-module, and set N=HomR(M, R) and 5= Homñ(M, M).Set (m,f) = (m)f for m e M,fe N; and [/, m] is defined via mx[f, m] = (m1,f)m for all m, w, e M,feN.Then (R, M, N, S) is a Morita context, called the standard context for RM.If R is semiprime and RM is torsionless, then the above condition is satisfied by the standard Morita context for RM.If RM is a generator and 1 e R; or indeed, if (Trace RM)m^0 whenever O^m e M, then the standard Morita context for RM satisfies the above condition.Lemma 1.(a) If A is an essential left ideal ofS, then MA is an essential submodule of RM.