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Journal Article

# Faithful Abelian Groups of Infinite Rank

Ulrich Albrecht
Proceedings of the American Mathematical Society
Vol. 103, No. 1 (May, 1988), pp. 21-26
DOI: 10.2307/2047520
Stable URL: http://www.jstor.org/stable/2047520
Page Count: 6

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Topics: Mathematical rings, Mathematical theorems, Isomorphism, Functors, Combinatorics

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## Abstract

Let $B$ be a subgroup of an abelian group $G$ such that $G/B$ is isomorphic to a direct sum of copies of an abelian group $A$. For $B$ to be a direct summand of $G$, it is necessary that $G$ be generated by $B$ and all homomorphic images of $A$ in $G$. However, if the functor $\operatorname{Hom}(A, -)$ preserves direct sums of copies of $A$, then this condition is sufficient too if and only if $M \otimes_{E(A)} A$ is nonzero for all nonzero right $E(A)$-modules $M$. Several examples and related results are given.

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