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.
1
Expert's answer
2013-02-06T08:35:25-0500
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.

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