Question #17648

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.

Expert's answer

Let *f *: *M → M *be asurjective (resp. injective) endomorphism of a finitely generated left *S*-module*M*. 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.

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