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On endomorphisms of multiplication modules. (English) Zbl 0820.13003

Let \(R\) be a commutative ring with identity. An \(R\)-module \(M\) is called a multiplication module if for every submodule \(N\) of \(M\) there exists an ideal \(I\) of \(R\) such that \(N= IM\). If \(M\) is a finitely generated multiplication module then every endomorphism of \(M\) is given by multiplication by an element of \(R\), and a partial converse is given for finitely generated modules. In addition, if \(M\) is an indecomposable multiplication module which satisfies dcc on multiplication submodules then any non-nilpotent endomorphism of \(M\) is an automorphism, and hence \(\text{End}(M)\) is a local ring.

MSC:

13A05 Divisibility and factorizations in commutative rings
13C10 Projective and free modules and ideals in commutative rings
13B10 Morphisms of commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C13 Other special types of modules and ideals in commutative rings
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