Invariants for a class of torsion-free abelian groups

Authors:
D. Arnold and C. Vinsonhaler

Journal:
Proc. Amer. Math. Soc. **105** (1989), 293-300

MSC:
Primary 20K15

DOI:
https://doi.org/10.1090/S0002-9939-1989-0935102-X

MathSciNet review:
935102

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Abstract: In this note we present a complete set of quasi-isomorphism invariants for strongly indecomposable abelian groups of the form . Here are subgroups of the rationals and is the kernel of , where . The invariants are the collection of numbers , where ranges over all subsets of the type lattice generated by . Our results generalize the classical result of Baer for finite rank completely decomposable groups, as well as a result of F. Richman on a subset of the groups of the form .

**[A-1]**David M. Arnold,*Finite rank torsion free abelian groups and rings*, Lecture Notes in Mathematics, vol. 931, Springer-Verlag, Berlin-New York, 1982. MR**665251****[A-2]**-,*Representations of partially ordered sets and abelian groups*, Proceedings of the 1987 Perth Conference on Abelian Groups (to appear).**[A-3]**David M. Arnold,*Pure subgroups of finite rank completely decomposable groups*, Abelian group theory (Oberwolfach, 1981) Lecture Notes in Math., vol. 874, Springer, Berlin-New York, 1981, pp. 1–31. MR**645913****[AV-1]**D. Arnold and C. Vinsonhaler,*Pure subgroups of finite rank completely decomposable groups. II*, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 97–143. MR**722614**, https://doi.org/10.1007/BFb0103698**[AV-2]**D. Arnold and C. Vinsonhaler,*Representing graphs for a class of torsion-free abelian groups*, Abelian group theory (Oberwolfach, 1985) Gordon and Breach, New York, 1987, pp. 309–332. MR**1011321****[AV-3]**D. Arnold and C. Vinsonhaler,*Quasi-isomorphism invariants for a class of torsion-free abelian groups*, Houston J. Math.**15**(1989), no. 3, 327–340. MR**1032393****[AV-4]**D. M. Arnold and C. I. Vinsonhaler,*Endomorphism rings of Butler groups*, J. Austral. Math. Soc. Ser. A**42**(1987), no. 3, 322–329. MR**877802****[AV-5]**-,*Coxeter functors and duality for abelian groups*, preprint.**[Ba]**Reinhold Baer,*Abelian groups without elements of finite order*, Duke Math. J.**3**(1937), no. 1, 68–122. MR**1545974**, https://doi.org/10.1215/S0012-7094-37-00308-9**[BB]**Sheila Brenner and M. C. R. Butler,*Endomorphism rings of vector spaces and torsion free abelian groups*, J. London Math. Soc.**40**(1965), 183–187. MR**0174593**, https://doi.org/10.1112/jlms/s1-40.1.183**[BP]**R. A. Beaumont and R. S. Pierce,*Torsion free groups of rank two*, Mem. Amer. Math. Soc. No.**38**(1961), 41. MR**0130297****[Bu-1]**M. C. R. Butler,*A class of torsion-free abelian groups of finite rank*, Proc. London Math. Soc. (3)**15**(1965), 680–698. MR**0218446**, https://doi.org/10.1112/plms/s3-15.1.680**[Bu-2]**M. C. R. Butler,*Torsion-free modules and diagrams of vector spaces*, Proc. London Math. Soc. (3)**18**(1968), 635–652. MR**0230767**, https://doi.org/10.1112/plms/s3-18.4.635**[Bu-3]**-,*Some almost split sequences in torsion-free abelian group theory*, Abelian Group Theory, Gordon and Breach, New York, 1987, pp. 291-302.**[Re]**James D. Reid,*On the ring of quasi-endomorphisms of a torsion-free group*, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman and Co., Chicago, Ill., 1963, pp. 51–68. MR**0169915****[R-1]**Fred Richman,*Butler groups, valuated vector spaces, and duality*, Rend. Sem. Mat. Univ. Padova**72**(1984), 13–19. MR**778329****[R-2]**Fred Richman,*An extension of the theory of completely decomposable torsion-free abelian groups*, Trans. Amer. Math. Soc.**279**(1983), no. 1, 175–185. MR**704608**, https://doi.org/10.1090/S0002-9947-1983-0704608-X

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0935102-X

Article copyright:
© Copyright 1989
American Mathematical Society