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Duality in some classes of torsion-free Abelian groups of finite rank. (English. Russian original) Zbl 0633.20031
Sib. Math. J. 27, 563-571 (1986); translation from Sib. Mat. Zh. 27, No. 4(158), 117-127 (1986).
Let \(\sigma\), \(\tau\) be a pair of types of torsion-free (abelian) groups of rank 1 which are determined by characteristics \((k_ p)\), \((m_ p)\) such that \(k_ p\leq m_ p\) for all primes \(p\). A torsion-free group \(A\) of finite rank \(n\) belongs to the class \(D^{\tau}_{\sigma}\) iff there exists a free subgroup \(J\) of rank \(n\) of \(A\) such that any \(p\)-primary component of \(A/J\) is a direct sum of \(n\) groups isomorphic either to \(Z(p^{k_ p})\) or to \(Z(p^{m_ p})\). For primes \(p\) such that \(k_ p<m_ p\) let \(s_ p(A)\), \(t_ p(A)\) denote the number of isomorphic summands so \((s_ p(A),t_ p(A))\) are determined up to an equivalence (similar to that of characteristics). It is easy to see that the type of any element of any group in \(D^{\tau}_{\sigma}\) is greater than or equal to \(\sigma\). A detailed study of groups \(A/p^{m_ p}A\) and of \(p\)-adic completions of \(A\) leads to a notion of duality in the class \(D^{\tau}_{\sigma}\), i.e. a contravariant functor \(-^*: D^{\tau}_{\sigma}\to D^{\tau}_{\sigma}\) (where \(D^{\tau}_{\sigma}\) is considered as the category of groups A with a fixed basis of \(J\) and with quasihomomorphisms as maps) such that 1) rank \(A^*=rank A\), 2) \(s_ p(A^*)=t_ p(A)\), \(t_ p(A^*)=s_ p(A)\) for almost all \(p\), 3) \(A^{**}=A\), \(\phi^{**}=\phi\) for all groups \(A\) and all quasihomomorphisms \(\phi\). Any two such dualities are equivalent. This duality is compared with those defined by Warfield and Arnold; e.g. it is shown that if \(m_ p=\infty\) implies \(k_ p=\infty\) then \(-^*=\operatorname{Hom}(-,R)\) where \(R\) is a torsion free group of rank 1 and of type \(\sigma+\tau\).
Reviewer: St.Balcerzyk

MSC:
20K15 Torsion-free groups, finite rank
20K40 Homological and categorical methods for abelian groups
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[1] L. Ya. Kulikov, ?Generalized primary groups,? Tr. Mosk. Mat. Obshch.,1, 247-326 (1952).
[2] A. A. Beaumont and R. S. Pierce, ?Torsion-free rings,? Illinois J. Math.,5, No. 1, 61-98 (1961). · Zbl 0108.03802
[3] R. Baer, ?Abelian groups without elements of finite order,? Duke Math. J.,3, No. 1, 68-122 (1937). · Zbl 0016.20303
[4] D. M. Arnold, ?A duality for torsion-free modules of finite rank over a discrete valuation ring,? Proc. London Math. Soc.,24, No. 3, 204-216 (1972). · Zbl 0237.13016
[5] D. M. Arnold, ?A duality for quotient divisible Abelian groups of finite rank,? Pac. J. Math.,42, No. 1, 11-15 (1972). · Zbl 0262.20062
[6] R. B. Warfield, ?Homomorphisms and duality for torsion-free groups,? Math. Z.,107, No. 1, 189-200 (1968). · Zbl 0169.03602
[7] L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, New York-London (1970). · Zbl 0209.05503
[8] F. Richman, ?A class of rank 2 torsion-free groups,? in: Studies on Abelian Groups, Paris (1968), pp. 327-333.
[9] C. E. Murley, ?The classification of certain classes of torsion-free Abelian groups,? Pac. J. Math.,40, No. 3, 647-665 (1972). · Zbl 0261.20045
[10] L. Ya. Kulikov, Algebra and Number Theory [in Russian], Vysshaya Shkola, Moscow (1979).
[11] M. Bourbaki, Algebra I, Hermann, Paris (1974), Chaps. I?III.
[12] A. A. Fomin, ?Tensor products of torsion-free Abelian groups,? Sib. Mat. Zh.,16, No. 5, 1071-1080 (1975).
[13] A. A. Fomin, ?Monogeneous ?-groups,? in: Proceedings 7th All-Union Symposium on Group Theory, Krasnoyarsk State Univ., 128 (1980).
[14] D. W. Dubois, ?Cohesive groups and p-adic integers,? Publ. Math. Debrecen,12, No. 1, 51-58 (1965). · Zbl 0136.29002
[15] A. A. Fomin, ?Abelian groups with free subgroups of infinite index and their endomorphism rings,? mat. Zametki,36, No. 2, 179-187 (1984).
[16] A. A. Fomin, ?Abelian groups with free pure subgroups,? in: Proceedings 9th All-Union Symposium on Group Theory, MGPI im. V. I. Lenina, Moscow (1984), p. 162. · Zbl 0571.20047
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