Abstract
A multiplication on an Abelian group G is a homomorphism μ: G ⊗ G → G. An Abelian group G with a multiplication on it is called a ring on the group G. R. A. Beaumont and D. A. Lawver have formulated the problem of studying semisimple groups. An Abelian group is said to be semisimple if there exists a semisimple associative ring on it. Semisimple groups are described in the class of vector Abelian nonmeasurable groups. It is also shown that if a set I is nonmeasurable, \( G=\coprod_{i\in I}{A}_i \) is a reduced vector Abelian group, and μ is a multiplication on G, then μ is determined by its restriction on the sum \( {\displaystyle \begin{array}{c}\oplus \\ {}i\in I\end{array}}{A}_i \) this statement is incorrect if the set I is measurable or the group G is not reduced.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 5, pp. 243–258, 2019.
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Kompantseva, E.I. Rings on Vector Abelian Groups. J Math Sci 259, 552–562 (2021). https://doi.org/10.1007/s10958-021-05646-2
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DOI: https://doi.org/10.1007/s10958-021-05646-2