Multiplications on mixed abelian groups
https://doi.org/10.22405/2226-8383-2019-20-1-212-221
Abstract
A multiplication on an abelian group G is a homomorphism $$\mu: G\otimes G\rightarrow G$$. An mixed abelian group G is called an MT-group if every multiplication on the torsion part of the group G can be extended uniquely to a multiplication on G. MT-groups have been studied in many articles on the theory of additive groups of rings, but their complete description has not yet been obtained. In this paper, a pure fully invariant subgroup $$G^*_\Lambda$$ is considered for an abelian MT-group G. One of the main properties of this subgroup is that $$\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$$ is a nil-ideal in every ring with the additive group G (here $$\Lambda (G)$$ is the set of all primes p, for which the p-primary component of G is non-zero). It is shown that for every MT-group G either $$G=G^*_\Lambda$$ or the quotient group $$G/G^*_\Lambda$$ is uncountable.
About the Author
Ekaterina Igorevna KompantsevaRussian Federation
doctor of engineering, professor, Professor, Department of algebra, Moscow state pedagogical University; Professor of the Department of probability theory and mathematical statistics, Financial University under the Government of the Russian Federation, Moscow.
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Review
For citations:
Kompantseva E.I. Multiplications on mixed abelian groups. Chebyshevskii Sbornik. 2019;20(1):212-221. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-1-212-221
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