Choi, Chang Woo Multiplication modules and endomorphisms. (English) Zbl 0876.13001 Math. J. Toyama Univ. 18, 1-8 (1995). Summary: In this note all rings are commutative with an identity and all modules are unital. Let \(R\) be a ring. 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 \(N=IM\). It is clear that every cyclic \(R\)-module is a multiplication module. Also, W. W. Smith [Can. J. Math. 21, 1057-1061 (1969; Zbl 0183.04001); theorem 1] showed that any projective ideal of \(R\) is a multiplication \(R\)-module. An endomorphism \(\varphi\) of \(M\) will be called trivial if there exists \(a\in R\) such that \(\varphi (m)= am\) for all \(m\in M\).In this paper we study the structure and properties of multiplication modules in terms of endomorphisms. We prove that over a von Neumann regular ring \(R\) the ring \(R\) is self-injective if and only if for every ideal \(J\) of \(R\), every \(R\)-endomorphism of \(J\) is trivial. Also, we give some conditions for a finitely generated module (submodule) to be a multiplication module. Cited in 4 Documents MSC: 13A05 Divisibility and factorizations in commutative rings 13B10 Morphisms of commutative rings 13C13 Other special types of modules and ideals in commutative rings Keywords:von Neumann regular ring; self-injective ring; multiplication modules; endomorphisms Citations:Zbl 0183.04001 PDFBibTeX XMLCite \textit{C. W. Choi}, Math. J. Toyama Univ. 18, 1--8 (1995; Zbl 0876.13001)