FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 1, PAGES 229-233

Abelian groups as endomorphic modules over their endomorphism ring

D. S. Chistyakov
O. V. Lyubimcev

Abstract

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Let $R$ be an associative ring with a unit and $N$ be a left $R$-module. The set $M$_R(N) = {f: N ® N | f(rx) = rf(x), r Î R, x Î N} is a near-ring with respect to the operations of addition and composition and contains the ring $E$_R(N) of all endomorphisms of the $R$-module $N$. The $R$-module $N$ is endomorphic if $M$_R(N) = E_R(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank $2$.

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Last modified: December 21, 2006