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M. A. Turmanov 《Journal of Mathematical Sciences》2006,137(6):5336-5345
Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented
as the equality in an appropriate factor category. 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, a topical problem in the Abelian group theory is the problem of investigation of pureness
in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules
over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module
E
(G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary
ring R with the identity element. Indeed, if
R
M is an arbitrary unitary left module and M
+ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M
+) under the canonical ring homomorphism R → E(M
+). Then it holds that if
E(M+)
N is a pure submodule in
E(M+)
M
+, then
R
N is a pure submodule in
R
M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free
groups of finite rank will be investigated.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238,
2004. 相似文献
3.
Smolobochkin A. V. Turmanov R. A. Gazizov A. S. Appazov N. O. Burilov A. R. Pudovik M. A. 《Russian Journal of General Chemistry》2019,89(10):2143-2146
Russian Journal of General Chemistry - The reactions of 1-(4,4-diethoxybutyl)ureas with diphenyl chlorophosphine in the presence of acetic acid afforded new... 相似文献
4.
Smolobochkin A. V. Turmanov R. A. Gazizov A. S. Kuznetsova E. A. Burilov A. R. Pudovik M. A. 《Russian Journal of Organic Chemistry》2020,56(6):1119-1121
Russian Journal of Organic Chemistry - The reaction of N-(4,4-diethoxybutyl)phosphamides with chloro(diphenyl)phosphine in chloroform in the presence of acetic acid gave previously unknown... 相似文献
5.
We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module E(G)
G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups. 相似文献
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