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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