Abelian Groups - László Fuchs [Springer]

5 Special Endomorphism Rings 637
seem to suggest that specific endomorphism rings are indeed special. Needless to
say, we are primarily interested in conventional ring properties.
It seems sensible at this point to stress that one should not have high expectations
of the interrelations between interesting group and important ring properties. We
keep emphasizing that significant group and ring properties seldom match: groups
whose endomorphism rings are of special interest in ring theory are few and far
between.
We are going to survey several cases of interest.
Elementary Properties We start assembling a few simple observations that will
simplify subsequent arguments. We will abbreviate E D End A.
(a) Suppose ˛ 2 E.Ifnj˛,then˛A nA, and if n˛ D 0,then˛A AŒn for n 2 N.
Indeed, if ˇ 2 E satisfies nˇ D ˛, then˛A D nˇA nA, andifn˛ D 0, then
n˛A D 0,so˛A AŒn.
(b) If End A D R 1 ˚ R 2 is a ring direct sum, then A D B 1 ˚ B 2 where R i Š End B i
and Hom.B i ; B j / D 0 for i ¤ j. This is a simple consequence of Sect. 1(D).
(c) If A has a sequence of direct decompositions
A D A 0 ˚˚A n ˚ C n with C n D A nC1 ˚ C nC1 .n