Associative rings on vector groups

An abelian group is called semisimple if it is the additive group of a semisimple ring. R. A. Beaumont and D. A. Lawver have formulated the description problem for semisimple groups. We consider vector semisimple groups in the present paper. Vector groups are direct products $\prod\limits_{i\in I}R_i$ of torsion free abelian groups $R_i\, (i\in I)$ of rank 1. The semisimple vector groups $\prod\limits_{i\in I} R_i$ are described in the present paper in the case where $I$ is a not greater than countable set.
A multiplication on an abelian group $G$ is a homomorphism $\mu\colon G\otimes G\rightarrow G$, we denote it as $\mu(g_1\otimes g_2)=g_1\times g_2$ for $g_1, g_2\in G$. The group $G$ with a multiplication $\times$ is called the ring on the group $G$ and it is denoted as $(G,\times)$. It is shown that every multiplication on a direct product of torsion free rank-1 groups is determined by its restriction on the direct sum of these groups. In particular, the following statement takes place.
Lemma 3. Let $I$ be a not greater than countable set, $G=\prod\limits_{i\in I}R_i$ and $S=\bigoplus\limits_{i\in I} R_i$. Let $\times$ be a multiplication on the group $G$. If the restriction of this multiplication on $S$ is zero, then the multiplication itself is zero.
Let $\prod\limits_{i\in I}R_i$ be a vector group. We use the following notations: $t(R_i)$ is the type of the group $R_i$, $I_0$ is the set of indices $i\in I$ such that $t(R_i)$ is an idempotent type with an infinite number of zero components. If $k\in I$, then $I_0(k)$ is the set of indices $i\in I_0$ such that $t(R_i)\geq t(R_k)$.
Theorem 1. Let $I$ be a not greater than countable set. A reduced vector group $\prod\limits_{i\in I} R_i$ is semisimple if and only if
1) there are no groups $R_i\, (i\in I)$ of an idempotent type, where the number of zero components is finite;
2) the set $I_0(k)$ is infinite for every group $R_k$ of the not idempotent type.
Note that the set of types of groups $R_i\,(i\in I)$ is an invariant of the group $G=\prod\limits_{i\in I} R_i$, if $I$ is a not greater than countable set. Therefore, this description doesn't depend on the decomposition of the group $G$ into a direct product of rank-1 groups.
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