I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2004, VOLUME 10, NUMBER 2, PAGES 225-238
On pureness in Abelian groups
M. A. Turmanov
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Torsion-free Abelian groups and are called
quasi-equal () if for a certain natural
It is known that the quasi-equality of torsion-free
Abelian groups can be represented as the equality in an appropriate
Thus while dealing with certain group properties it is usual to prove
that the property under consideration is preserved under the
transition to a quasi-equal group.
This trick is especially frequently used when the author investigates
module properties of Abelian groups, here a group is considered
as a left module over its endomorphism ring.
On the other hand, an actual problem in the Abelian group theory is
a problem of investigation of pureness in the category of Abelian
We consider the pureness introduced by P. Cohn for
Abelian groups as modules over their endomorphism rings.
The feature of the investigation of the properties of pureness for the
Abelian group as the module lies in
the fact that this is a more general situation than the
investigation of pureness for a unitary module over an arbitrary
with the identity element.
Indeed, if is an
arbitrary unitary left module and is its
Abelian group, then each element from can be identified with an
appropriate endomorphism from the ring under the
canonical ring homomorphism .
Then it holds that if
is a pure submodule in , then
is a pure submodule in .
In the present paper the interrelations between pureness, servantness,
and quasi-decompositions for Abelian torsion-free groups of finite
rank will be investigated.
Last modified: December 23, 2004