Answer to Question #23563 in Algebra for Mohammad

Question

Let R be a finite-dimensional k-algebra, M be an R-module and E = EndRM. Show that if f ∈ E is such that f(M) ⊆ (rad R)M, then f ∈ rad E.

Answer 1

Let I = {f ∈ E : f(M) ⊆(rad R)M}. It is routine to check that I is anideal in the endomorphism ring E. For this ideal I, we have IⁿM⊆(rad R)ⁿM. Since rad R is nilpotent, IⁿM= 0 for a sufficiently large n, so Iⁿ = 0. Thisimplies that I ⊆rad E, as desired.