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Extension functors on the category of \(A\)-solvable abelian groups. (English) Zbl 0776.20018
Let \(A\) and \(G\) be abelian groups. The group \(H_ A(G)=\operatorname{Hom}(A,G)\) has an \(E(A)\)-module structure in the natural way. Conversely, we set \(T_ A(M)= M\otimes_{E(A)}A\) for any right \(E(A)\)-module \(M\). The group \(G\) is said to be \(A\)-solvable, if the natural evaluation map \(\theta_ G: T_ A H_ A(G)\to G\) is an isomorphism. The author investigates some extension functors on the category \(C_ A\) of \(A\)-solvable groups. The methods developed are used for a construction of \(A\)-solvable groups in the case that \(A\) is slender and has rank at least two.
Reviewer: L.Bican (Praha)

MSC:
20K40 Homological and categorical methods for abelian groups
16S50 Endomorphism rings; matrix rings
20K35 Extensions of abelian groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
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References:
[1] Albrecht U.: Endomorphism rings and \(A\)-projective torsion-free groups. Abelian Group Theory, Honolulu 1983, Springer LNM 1006 (1983); 209-227.
[2] Albrecht U.: Baer’s Lemma and Fuchs’ Problem 84a. Trans. Amer. Math. Soc. 293 (1986); 565-582. · Zbl 0592.20058 · doi:10.2307/2000022
[3] Albrecht U.: Faithful abelian groups of infinite rank. Proc. Amer. Math. Soc. 103 (1988); 21-26. · Zbl 0646.20042 · doi:10.2307/2047520
[4] Albrecht U.: Abelian groups, \(A\), such that the category of \(A\)-solvabIe groups is preabelian. Abelian Group Theory, Perth 1987; Contemporary Mathematics, Vol. 87; American Mathematical Society; Providence (1987); 117-132. · doi:10.1090/conm/087/995270
[5] Albrecht U.: Endomorphism rings of faithfully flat abelian groups. to appear in Resultate der Mathematik. · Zbl 0709.20031 · doi:10.1007/BF03322457
[6] Arnold D., Lady I..: Endomorphism rings and direct sums of torsion-free abelian groups. Trans. Amer. Math. Soc. 211 (1975); 225-237. · Zbl 0329.20033 · doi:10.2307/1997231
[7] Arnold D., Murley C.: Abelian groups, \(A\) such that \(\operatorname{Hom}(A,-)\) preserves direct sums of copies of \(A\). Pac. J. of Math. 56 (1975); 7-20. · Zbl 0337.13010 · doi:10.2140/pjm.1975.56.7
[8] Dugas M., Göbel R.: Every cotorsion-free ring is an endomorphism ring. Proc. London Math. Soc. 45 (1982); 319-336. · Zbl 0506.16022 · doi:10.1112/plms/s3-45.2.319
[9] Fuchs L.: Infinite Abelian Groups. Vol. I and II, Academic Press; London, New York (1970/73). · Zbl 0209.05503
[10] Jans J.: Rings and Homology. Reinhold-Winston; New York (1979). · Zbl 0141.02901
[11] MacLane S.: Homology. Academic Press; London, New York (1963). · Zbl 0133.26502
[12] Rotman J.: An Introduction to Homological Algebra. Academic Press; London, New York (1982). · Zbl 1157.18001 · doi:10.1007/b98977
[13] Richman F., Walker E.: Ext in pre-abelian categories. Pac. J. of Math. 71 (2) (1977); 521-535. · Zbl 0354.18018 · doi:10.2140/pjm.1977.71.521
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