When is Ext(A,B) torsion-free?, and related problems. (English) Zbl 0653.20055

This paper considers the problem: Given torsion-free abelian groups A and B, when is Ext(A,B) torsion-free? The main result is the following: Theorem. Let A be countable, let B be finite rank with outer type of B not equal to the type of Q, the rational numbers, and let X be a subgroup of Q of type equal to the inner type of B. Then Ext(A,B) is torsion-free if and only if \(\cap \{Ker f|\) f: \(A\to X\}\) is divisible by all primes p such that pX\(\neq X.\)
This theorem extends an earlier result of Huber and Warfield for the case when B is rank one. The author also shows that under the hypotheses of the theorem, if Ext(A,B) is torsion-free, then there is a projective- injective relationship between A and B. Specifically, (1) B is injective with respect to any pure embedding \(0\to C\to A\) where C is finite rank, and (2) A if projective with respect to any epimorphism \(B\to C\to 0\) where C is torsion-free.
Reviewer: C.Vinsonhaler


20K15 Torsion-free groups, finite rank
20K35 Extensions of abelian groups
20K40 Homological and categorical methods for abelian groups
Full Text: DOI


[1] Albrecht, U. 1982/1983. Endomorphism Rings and A-projective. Abelian Group Theory Proceedings Honolulu. 1982/1983. Vol. LNM1006, Springer-Verlag.
[2] Arnold D.M., Finite Rank Torsion-Free Abelian Groups and Rings 931 (1980)
[3] Arnold D.M., Rocky Mountain Journal 12 (1982)
[4] DOI: 10.1112/plms/s3-38.3.532 · Zbl 0406.20044
[5] Faticoni T., Ph.D. thesis (1981)
[6] Faticoni T., On Torsion-Free Ext
[7] Fuchs L., Infinite Abelian Groups 1 (1970) · Zbl 0209.05503
[8] Fuchs L., Infinite Abelian Groups 2 (1973) · Zbl 0257.20035
[9] Goeters H.P., Generating Cotorsion Theories and Injective Classes · Zbl 0643.20033
[10] Goeters H.P., Generating an Injective Class of Torsion-Free Abelian Groups · Zbl 0681.20035
[11] Goeters H.P., Rocky Mountain Journal
[12] Goeters H. P., On the p-ranks of Hom(A,B) · Zbl 0683.20040
[13] Huber, M. and Warfield, R.B. 1981. Homomorphlsms Between Cartesian Powers of an Abellan Groups. AbelIan Group Theory Proceedings Oberwolfach. 1981. Vol. LNM 874, Springer-Verlag.
[14] DOI: 10.1007/BF01238460 · Zbl 0402.20043
[15] Rotman J., An Introduction to Homological Algebra (1979) · Zbl 0441.18018
[16] Vinsonhaler, C. and Wickless, W. 1982.Projective Classes of Abelian Groups, Vol. 39, 195–215. Hungar: Acta. Math Acad. Sci. · Zbl 0496.20041
[17] Vinsonhaler, C. and Wickless, W. 1982/1983. Projective and Injective Classes of Completely Decomposable Groups. Abelian Group Theory Proceedings Honolulu. 1982/1983. Vol. LNM1006, Springer-Verlag. · Zbl 0496.20041
[18] Walker C., Acta Math 23 pp 145– (1972)
[19] DOI: 10.1007/BF01304857 · Zbl 0244.20064
[20] DOI: 10.1007/BF01110257 · Zbl 0169.03602
[21] Wickless, W. 1984.Projective Classes of Torsion-Free Abelian Groups II, Vol. 43, 13–20. Hungar: Acta. Math Acad. Sci. · Zbl 0547.20051
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