Classification theory of abelian groups. I. Balanced projectives

Warfield, R. B.

doi:10.1090/S0002-9947-1976-0422455-X

Classification theory of abelian groups. I. Balanced projectives
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Trans. Amer. Math. Soc. 222 (1976), 33-63 Request permission

Abstract:

We introduce in this paper a class of Abelian groups which includes the torsion totally projective groups and those torsion-free groups which are direct sums of groups of rank one. Characterizations of the groups in this class are given, and a complete classification theorem, in terms of additive numerical invariants, is proved.

References

D. M. Arnold and E. L. Lady, Endomorphism rings and direct sums of torsion free abelian groups, Trans. Amer. Math. Soc. 211 (1975), 225–237. MR 417314, DOI 10.1090/S0002-9947-1975-0417314-1
Reinhold Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68–122. MR 1545974, DOI 10.1215/S0012-7094-37-00308-9
Bernard Charles, Sous-groupes fonctoriels et topologies, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 75–92 (French). MR 0240195
Peter Crawley and Alfred W. Hales, The structure of abelian $p$-groups given by certain presentations, J. Algebra 12 (1969), 10–23. MR 238947, DOI 10.1016/0021-8693(69)90014-3
L. Fuchs, Notes on abelian groups. I, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 2 (1959), 5–23. MR 123608
László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673

On the classification of Abelian groups

Irving Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. MR 0065561
Irving Kaplansky, Projective modules, Ann. of Math (2) 68 (1958), 372–377. MR 0100017, DOI 10.2307/1970252
Irving Kaplansky and George W. Mackey, A generalization of Ulm’s theorem, Summa Brasil. Math. 2 (1951), 195–202. MR 49165
George Kolettis Jr., Direct sums of countable groups, Duke Math. J. 27 (1960), 111–125. MR 110748
George Kolettis, Homogeneously decomposable modules, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 223–238. MR 0244228
L. Ya. Kulikov, On direct decompositions of groups, Ukrain. Mat. Žurnal 4 (1952), 230–275, 347–372 (Russian). MR 0058598
L. Ya. Kulikov, Generalized primary groups. I, Trudy Moskov. Mat. Obšč. 1 (1952), 247–326 (Russian). MR 0049188
C. E. Murley, Direct products and sums of torsion-free Abelian groups, Proc. Amer. Math. Soc. 38 (1973), 235–241. MR 311800, DOI 10.1090/S0002-9939-1973-0311800-4
R. J. Nunke, Purity and subfunctors of the identity, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman & Co., Chicago, Ill., 1963, pp. 121–171. MR 0169913
R. J. Nunke, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212. MR 218452, DOI 10.1007/BF01135839
Larry D. Parker and Elbert A. Walker, An extension of the Ulm-Kolettis theorems, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 309–325. MR 0265458
Joseph Rotman and Ti Yen, Modules over a complete discrete valuation ring, Trans. Amer. Math. Soc. 98 (1961), 242–254. MR 122895, DOI 10.1090/S0002-9947-1961-0122895-1
A. E. Stratton, Mixed modules over an incomplete discrete valuation ring, Proc. London Math. Soc. (3) 21 (1970), 201–218. MR 272772, DOI 10.1112/plms/s3-21.2.201
Helmut Ulm, Zur Theorie der abzählbar-unendlichen Abelschen Gruppen, Math. Ann. 107 (1933), no. 1, 774–803 (German). MR 1512826, DOI 10.1007/BF01448919
C. L. Walker and R. B. Warfield Jr., Unique decomposition and isomorphic refinement theorems in additive categories, J. Pure Appl. Algebra 7 (1976), no. 3, 347–359. MR 414657, DOI 10.1016/0022-4049(76)90059-1
Elbert A. Walker, Ulm’s theorem for totally projective groups, Proc. Amer. Math. Soc. 37 (1973), 387–392. MR 311805, DOI 10.1090/S0002-9939-1973-0311805-3
R. B. Warfield Jr., Homomorphisms and duality for torsion-free groups, Math. Z. 107 (1968), 189–200. MR 237642, DOI 10.1007/BF01110257
R. B. Warfield Jr., Classification theorems for $p$-groups and modules over a discrete valuation ring, Bull. Amer. Math. Soc. 78 (1972), 88–92. MR 291284, DOI 10.1090/S0002-9904-1972-12870-2

Simply presented groups

R. B. Warfield Jr., The uniqueness of elongations of Abelian groups, Pacific J. Math. 52 (1974), 289–304. MR 360866, DOI 10.2140/pjm.1974.52.289
R. B. Warfield Jr., A classification theorem for abelian $p$-groups, Trans. Amer. Math. Soc. 210 (1975), 149–168. MR 372071, DOI 10.1090/S0002-9947-1975-0372071-2

Classification theorems for infinite Abelian groups