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Fundam. Prikl. Mat., 2019, Volume 22, Issue 5, Pages 243–258 (Mi fpm1850)  

Rings on vector Abelian groups

E. I. Kompantsevaab

a Financial University under the Government of the Russian Federation, Moscow
b Moscow State Pedagogical University

Abstract: A multiplication on an Abelian group $G$ is a homomorphism $\mu\colon G\otimes G\rightarrow G$. 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, $G=\prod\limits_{i \in I} A_i$ is a reduced vector Abelian group, and $\mu$ is a multiplication on $G$, then $\mu$ is determined by its restriction on the sum $\bigoplus\limits_{i\in I} A_i$; this statement is incorrect if the set $I$ is measurable or the group $G$ is not reduced.

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UDC: 512.541

Citation: E. I. Kompantseva, “Rings on vector Abelian groups”, Fundam. Prikl. Mat., 22:5 (2019), 243–258

Citation in format AMSBIB
\Bibitem{Kom19}
\by E.~I.~Kompantseva
\paper Rings on vector Abelian groups
\jour Fundam. Prikl. Mat.
\yr 2019
\vol 22
\issue 5
\pages 243--258
\mathnet{http://mi.mathnet.ru/fpm1850}


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