# Answer to Question #23563 in Algebra for Mohammad

Question #23563

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.

Expert's answer

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*I*^{n}M*⊆*(rad*R*)*. Since rad*^{n}M*R*is nilpotent,*I*= 0 for a sufficiently large^{n}M*n*, so*I*= 0. Thisimplies that^{n}*I**⊆*rad*E*, as desired.Need a fast expert's response?

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