Answer to Question #17190 in Algebra for Hym@n B@ss

Question

Show that a nonzero ring R is simple iff each simple left Rmodule is faithful

Answer 1

First assume R is simple andlet M be a simple left R-module. Then ann(M) <>R. Since ann(M) is an ideal, we must have ann(M) = 0, so Mis faithful. Conversely, assume that R is not simple. Then R hasan ideal I <> 0,R. Let m be a maximal left ideal containingI and letM = R/m. Then RM is simple, but it is notfaithful since IM = 0.