Abstract
Let R be a commutative ring and M an R-module. Then M is a multiplication module if N = (N : M) M for each submodule N of M. The ideal theta(M) = Sigma(mis an element ofM)(Rm : M) of R has proved useful in studying multiplication modules. We show that if M is a faithful multiplication module, then theta(M) = boolean AND {I an ideal of R \ IM = M} = tau(M), the trace ideal of M. Moreover, theta(M) is an idempotent multiplication ideal of R and theta(theta(M)) = theta(M). We also show that for a multiplication module M, theta(M) / (0 : M) is an ideal of the endomorphism ring End(M-R) of M and that End(M-R) approximate to (lim) under left arrow (R/(0: N)) where the inverse limit is taken over the finitely generated submodules N of M.