Rings on vector Abelian groups

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.