Injective Endomorphisms of Finitely Generated Modules

Abstract

Let R be a commutative ring. Then any injective endomorphism of a finitely generated R-module is always an isomorphism if and only if R is 0-dimensional, that is, if every prime ideal is maximal. This note aims at considering cases where an injective endomorphism of a finitely generated module is, actually, an isomorphism. It is a simple exercise that artinian modules are endowed with this property [1, p. 23] and here we will show that the commutative rings for which the fact above is always true resemble artinian rings. A similar question on when surjective endomorphisms of finitely generated modules are isomorphisms was proved independently by Strooker [3] and the author [4] or [5] for any commutative ring, regardless of finite presentation [2, p. 35 ] or chain conditions [1, p. 23]. For a commutative ring R, the result of this note says THEOREM. Any injective endomorphism of a finitely generated Rmodule is an isomorphism if and only if every prime ideal of R is maximal. That the above condition is necessary, it is easy to see: If PCQ are two distinct primes in R, then any element xEQ-P induces, via multiplication, an injection of R/P which is not surjective. The converse takes longer to prove but it is just as easy. Consider thus a ring R with the aforementioned property, that is, of having Krull dimension 0, and let f be an injection of the finitely generated R-module M. This module can be made into an R[x]module by defining x m=f(m) for m M. We claim that as an R [x ]-module M has an annihilator I, big enough, so that S = R [x ]/I is 0-dimensional. To see this, let ml, * * *, m1, be a generating set for M as an R-module. We have xmi= Ermi,m with rijCR, that is, a system of equations Received by the editors February 6, 1970. AMS Subject Classifications. Primary 1320; Secondary 1340.