Answer to Question #25258 in Abstract Algebra for Tsit Lam

Question

Let R be a ring which acts faithfully and irreducibly on a left module V . Let v ∈ V and A be a nonzero right ideal in R. Show that A • v = 0 ⇒ v = 0.

Answer 1

If v is nonzero, then we must have Rv = V .Since A · (Rv)=(AR) · v = A · v =0, the faithfulness of V implies that A = 0, a
contradiction.