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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Baer's lemma and Fuchs's problem 84a


Author: Ulrich Albrecht
Journal: Trans. Amer. Math. Soc. 293 (1986), 565-582
MSC: Primary 20K20; Secondary 16A50, 16A65, 20K30
DOI: https://doi.org/10.1090/S0002-9947-1986-0816310-7
MathSciNet review: 816310
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Abstract: An indecomposable, torsion-free, reduced abelian group $ A$ has the properties that (i) each subgroup $ B$ of an $ A$-projective group with $ {S_A}(B) = B$ is $ A$-projective and (ii) each subgroup $ B$ of a group $ G$ with $ {S_A}(G) + B = G$ and $ G/B$ $ A$-projective is a direct summand if and only if $ A$ is self-small and flat as a left $ E(A)$-module, and $ E(A)$ is right hereditary. Furthermore, a group-theoretic characterization is given for torsion-free, reduced abelian groups with a right and left Noetherian, hereditary endomorphism ring. This is applied to Fuchs' Problem 84a. Finally, various applications of the results of this paper are given.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0816310-7
Keywords: Endomorphism ring, $ A$-projective group, Baer's Lemma, two-sided Noetherian, hereditary ring, nonsingular finitely generated module
Article copyright: © Copyright 1986 American Mathematical Society