Question

Let ϕ : R → S be a ring homomorphism such that S is finitely generated when it is viewed as a left R-module via ϕ. If, over R, all finitely generated left modules are hopfian (resp. cohopfian), show that the same property holds over S.

Answer 1

Let f : M → M be asurjective (resp. injective) endomorphism of a finitely generated left S-moduleM. Via ϕ, we may view M as a left R-module,and, since RS is finitely generated, so is RM. Viewing f :M → M as a surjective (resp. injective) R-homomorphism, we inferfrom the assumption on R that f is an R-isomorphism, andhence an S-isomorphism.