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Classification theory of abelian groups. I. Balanced projectives


Author: R. B. Warfield
Journal: Trans. Amer. Math. Soc. 222 (1976), 33-63
MSC: Primary 20K40
DOI: https://doi.org/10.1090/S0002-9947-1976-0422455-X
MathSciNet review: 0422455
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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.


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  • [1] D. Arnold and E. Lady, Endomorphism rings and direct sums of torsion-free Abelian groups, Trans. Amer. Math. Soc. 211 (1975), 225-237. MR 0417314 (54:5370)
  • [2] R. Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), 68-122. MR 1545974
  • [3] B. Charles, Sous-groupes fonctoriels et topologíes, Studies on Abelian Groups (Sympos., Montpellier, 1967), Springer-Verlag, Berlin and New York; Dunod, Paris, 1968, pp. 75-92. MR 39 # 1547. MR 0240195 (39:1547)
  • [4] P. Crawley and A. W. Hales, The structure of Abelian p-groups given by certain presentations, J. Algebra 12 (1969), 10-23. MR 39 # 307. MR 0238947 (39:307)
  • [5] L. Fuchs, Notes on Abelian groups. I, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 2 (1959), 5-23. MR 23 # A933. MR 0123608 (23:A933)
  • [6] -, Infinite Abelian groups, Vols. I, II, Academic Press, New York, 1970, 1973. MR 41 # 333; 50 # 2362. MR 0255673 (41:333)
  • [7] P. Hill, On the classification of Abelian groups, Lecture notes.
  • [8] I. Kaplansky, Infinite Abelian groups, Univ. of Michigan Press, Ann Arbor, Mich., 1954; rev. ed., 1969. MR 16, 444; 38 # 2208. MR 0065561 (16:444g)
  • [9] -, Projective modules, Ann. of Math. (2) 68 (1958), 372-377. MR 20 # 6453. MR 0100017 (20:6453)
  • [10] I. Kaplansky and G. W. Mackey, A generalization of Ulm's theorem, Summa Brasil. Math. 2 (1951), 195-202. MR 14, 128. MR 0049165 (14:128b)
  • [11] G. Kolettis, Direct sums of countable groups, Duke Math. J. 27 (1960), 111-125. MR 22 # 1616. MR 0110748 (22:1616)
  • [12] -, Homogeneously decomposable modules, Studies on Abelian Groups (Sympos., Montpellier, 1967), Springer-Verlag, Berlin and New York; Dunod, Paris, 1968, pp. 223-238. MR 39 # 5545. MR 0244228 (39:5545)
  • [13] L. Kulikov, On direct decompositions of groups, Ukrain. Mat. Ž. 4 (1952), 230-275, 347-372; English transl., Amer. Math. Soc. Transl. (2) 2 (1956), 23-87. MR 15, 396; 17, 824. MR 0058598 (15:396c)
  • [14] -, Generalized primary groups. I, II, Trudy Moskov. Mat. Obšč. 1 (1952), 247-326; ibid. 2 (1953), 85-167. (Russian) MR 14, 132; 15, 9. MR 0049188 (14:132d)
  • [15] C. Murley, Direct products and sums of torsion-free Abelian groups, Proc. Amer. Math. Soc. 38 (1973), 235-241. MR 47 # 362. MR 0311800 (47:362)
  • [16] R. Nunke, Purity and subfunctors of the identity, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman, Chicago, 1963, pp. 121-171. MR 30 # 156. MR 0169913 (30:156)
  • [17] -, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182-212. MR 35 # 5508. MR 0218452 (36:1538)
  • [18] L. D. Parker and E. A. Walker, An extension of the Ulm-Kolettis theorems, Studies on Abelian Groups (Sympos., Montpellier, 1967), Springer-Verlag, Berlin and New York; Dunod, Paris, 1968, pp. 309-325. MR 42 # 367. MR 0265458 (42:367)
  • [19] J. Rotman and Ti Yen, Modules over a complete discrete valuation ring, Trans. Amer. Math. Soc. 98 (1961), 242-254. MR 23 # A227. MR 0122895 (23:A227)
  • [20] A. E. Stratton, Mixed modules over an incomplete discrete valuation ring, Proc. London Math. Soc. 21 (1970), 201-218. MR 0272772 (42:7653)
  • [21] H. Ulm, Zur Theorie der abzahlbar-unendlichen abelschen Gruppen, Math. Ann. 107 (1933), 774-803. MR 1512826
  • [22] C. Walker and R. B. Warfield, Jr., Unique decomposition and isomorphic refinement theorems in additive categories, J. Pure Appl. Algebra 7 (1976), 347-359. MR 0414657 (54:2756)
  • [23] E. A. Walker, Ulm's theorem for totally projective groups, Proc. Amer. Math. Soc. 37 (1973), 387-392. MR 47 # 367. MR 0311805 (47:367)
  • [24] R. B. Warfield, Jr., Homomorphisms and duality for torsion-free groups, Math. Z. 107 (1968), 189-200. MR 38 # 5923. MR 0237642 (38:5923)
  • [25] 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 45 # 378. MR 0291284 (45:378)
  • [26] -, Simply presented groups, Proc. Sem. Abelian Group Theory, Univ. of Arizona lecture notes, 1972.
  • [27] -, The uniqueness of elongations of Abelian groups, Pacific J. Math. 52 (1974), 289-304. MR 50 # 13313. MR 0360866 (50:13313)
  • [28] -, A classification theorem for Abelian p-groups, Trans. Amer. Math. Soc. 210 (1975), 149-168. MR 0372071 (51:8288)
  • [29] B. Wick, Classification theorems for infinite Abelian groups, Dissertation, Univ. of Washington, 1972.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0422455-X
Keywords: Ulm's theorem, totally projective groups, completely decomposable groups, invariants for Abelian groups, modules over a discrete valuation ring, extension of homomorphisms and isomorphisms, relative projectives, additive categories, decompositions of groups
Article copyright: © Copyright 1976 American Mathematical Society

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