Answer to Question #17274 in Algebra for sanches

Question

Show that any maximal ideal I of R is the annihilator of some simple left R-module, but not conversely.

Answer 1

Let V = R/m, where mis a maximal left ideal of R containing I. Then IV = 0, soI ⊆ ann(V ). Since ann(V ) is an ideal of R, themaximality of I implies that I = ann(V ).