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

View as HTML
View as gif image

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 $2$.

Location: http://mech.math.msu.su/~fpm/eng/k07/k071/k07114h.htm

Last modified: December 21, 2006