Abstract
Baer’s Lemma [9, Proposition 86.5] proved to be a useful tool for discussing decompositions of torsion-free abelian groups. If for abelian groups A and G the A-socle SA(G) of G is the subgroup of G generated by all f(A) where f ∈ HomZ(A,G) and G is A-projective if it is isomorphic to a direct summand of ⊕I A, then Baer’s Lemma can be formulated in the following way without using types.
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References
Albrecht, U.; Ideal conditions in endomorphism rings; Ph.D. thesis at New Mexico State University; Las Cruces (1982).
Arnold, D.M.; Finite Rank Torsion-free Abelian Groups and Rings; Springer Lecture Notes 931; Berlin, New York, Heidleberg (1982).
Arnold, D.M. and Lady, L.; Endomorphism rings and direct sums of torsion-free abelian groups; Trans. Amer. Math. Soc. 211 (1975); 225–237.
Arnold, D.M. and Murley, C.E.; Abelian groups A, such that Hom (A,-) preserves direct sums of copies of A; Pac. Journal of Math. 56 (1975), 7–20.
Bass, H.; Finistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.
Chatters, A. and Hajarnavis, C.; Rings with Chain Conditions; Pitman Advanced Publishing Program 44; London, Melbourne (1980).
Dugas, M. and Göbel, R.; Die Struktur kartesischer Produkte ganzer Zahlen modulo Produkte ganzer Zahlen; Math. Z. (1979); 15–21.
Fuchs, L.; Infinite Abelian Groups, Vol. I; Academic Press; New York (1970).
Fuchs, L.; Infinite Abelian Groups, Vol. II; Academic Press; New York (1973).
Göbel, R.; ZJberabzählbare abelsche Gruppen; Westdeutscher Verlag Oldenburg; to appear 1983.
Huber, M. and Warfield R.; Homomorphisms between cartesian powers of abelian groups; Abelian group Theory, Oberwohlfach 1981; Springer Lecture Notes 874; Berlin, Heidelberg, New York (1981); 202–227.
Rotman, J.; Notes on Homological Algebra; van Nostrand, Reinhold Mathematical Studies No. 26; van Nostrand-Reinhold Company; New York (1970).
Specker, E.; Additive Gruppen von Folgen ganzer Zahlen; Portugal. Math. 9 (1950); 131–140.
Walters, R.; Torsion-free reflexive modules of finite rank; Ph.D. Thesis at New Mexico State University, Las Cruces (1975).
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Albrecht, U. (1983). Endomorphism Rings and A-Projective Torsion-free Abelian Groups. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_9
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DOI: https://doi.org/10.1007/978-3-662-21560-9_9
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