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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Faithful abelian groups of infinite rank


Author: Ulrich Albrecht
Journal: Proc. Amer. Math. Soc. 103 (1988), 21-26
MSC: Primary 20K20; Secondary 20K30
DOI: https://doi.org/10.1090/S0002-9939-1988-0938637-8
MathSciNet review: 938637
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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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DOI: https://doi.org/10.1090/S0002-9939-1988-0938637-8
Keywords: Endomorphism ring, faithful, Baer's lemma
Article copyright: © Copyright 1988 American Mathematical Society