Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Global Warfield groups


Authors: Roger Hunter and Fred Richman
Journal: Trans. Amer. Math. Soc. 266 (1981), 555-572
MSC: Primary 20K21
DOI: https://doi.org/10.1090/S0002-9947-1981-0617551-X
MathSciNet review: 617551
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A global Warfield group is a summand of a simply presented abelian group. The theory of global Warfield groups encompasses both the theory of totally projective $ p$-groups, which includes the classical Ulm-Zippin theory of countable $ p$-groups, and the theory of completely decomposable torsion-free groups. This paper develops the central results of the theory including existence and uniqueness theorems. In addition it is shown that every decomposition basis of a global Warfield group has a nice subordinate with simply presented torsion cokernel, and that every global Warfield group is a direct sum of a group of countable torsion-free rank and a simply presented group.


References [Enhancements On Off] (What's this?)

  • [AHR] D. Arnold, R. Hunter and F. Richman, Global Azumaya theorems in additive categories, J. Pure Appl. Algebra 16 (1980), 223-242. MR 558485 (81j:18014)
  • [AHW] D. Arnold, R. Hunter and E. Walker, Direct sums of cyclic valuated groups, Sympos. Math. 23 (1979), 77-84. MR 565600 (81h:20066)
  • [CRH] P. Crawley and A. W. Hales, The structure of torsion abelian groups, Bull. Amer. Math. Soc. 74 (1968), 954-956. MR 0232840 (38:1163)
  • [FUC] L. Fuchs, Infinite Abelian groups, vol. II, Academic Press, New York, 1973. MR 0349869 (50:2362)
  • [HUN] R. Hunter, Balanced subgroups of abelian groups, Trans. Amer. Math. Soc. 215 (1976), 81-98. MR 0507068 (58:22337)
  • [HRW1] R. Hunter, F. Richman and E. Walker, Simply presented valuated abelian $ p$-groups, J. Algebra 49 (1977), 125-133. MR 0507069 (58:22338)
  • [HRW2] -, Existence theorems for Warfield groups, Trans. Amer. Math. Soc. 235 (1978), 345-362. MR 0473044 (57:12723)
  • [HRW3] -, Warfield modules, Abelian Group Theory (Proc. 2nd New Mexico State Univ. Conf., 1976), Lecture Notes in Math., vol. 616, Springer-Verlag, Berlin and New York, 1977, pp. 87-123. MR 0506216 (58:22041)
  • [KMA] I. Kaplansky and G. Mackey, A generalization of Ulm's theorem, Summa Brasil. Math. 2 (1951), 195-202. MR 0049165 (14:128b)
  • [MEG1] C. Megibben, On mixed groups of torsion-free rank one, Illinois J. Math. 11 (1967), 134-144. MR 0202832 (34:2691)
  • [MEG2] -, Modules over an incomplete discrete valuation ring, Proc. Amer. Math. Soc. 19 (1968), 450-452. MR 0222071 (36:5123)
  • [RIC] F. Richman, The constructive theory of $ KT$-modules, Pacific J. Math. 61 (1975), 621-637.
  • [RWA] F. Richman and E. Walker, Valuated groups, J. Algebra 56 (1979), 145-167. MR 527162 (80k:20053)
  • [ROT1] J. Rotman, Mixed modules over valuation rings, Pacific J. Math. 10 (1960), 607-623. MR 0114814 (22:5632)
  • [ROT2] -, Torsion-free and mixed abelian groups, Illinois J. Math. 5 (1961), 131-143. MR 0130908 (24:A762)
  • [RYN] J. Rotman and T. Yen, Modules over a complete discrete valuation ring, Trans. Amer. Math. Soc. 98 (1961), 242-254. MR 0122895 (23:A227)
  • [STA1] R. O. Stanton, An invariant for modules over a discrete valuation ring, Proc. Amer. Math. Soc. 49 (1975), 51-54. MR 0360572 (50:13020)
  • [STA2] -, Relative $ S$-invariants, preprint.
  • [STA3] -, Decomposition bases and Ulm's theorem, Abelian Group Theory (Proc. 2nd New Mexico State Univ. Conf., 1976), Lecture Notes in Math., vol. 616, Springer-Verlag, Berlin and New York, 1977, pp. 39-56. MR 0498902 (58:16913)
  • [STA4] Infinite decomposition bases, Pacific J. Math. (to appear). MR 0507066 (58:22335)
  • [STA5] -, Decomposition of modules over a discrete valuation ring, J. Austrialian Math. Soc. (to appear).
  • [STA6] -, Almost affable abelian groups, J. Pure Appl. Algebra (to appear). MR 532962 (80i:20029)
  • [STA7] -, $ S$-groups, preprint.
  • [STA8] -, Warfield groups and $ S$-groups, preprint.
  • [WAK] E. Walker, Ulm's theorem for totally projective groups, Proc. Amer. Math. Soc. 37 (1973), 387-392. MR 0311805 (47:367)
  • [WAL] K. Wallace, On mixed groups of torsion-free rank one with totally projective primary components, J. Algebra 17 (1971), 482-488. MR 0272891 (42:7772)
  • [WAR1] R. B. Warfield, Jr., Invariants and a classification theorem for modules over a discrete valuation ring, Univ. of Washington notes, 1971.
  • [WAR2] -, Classification theorems for $ p$-groups and modules over a discrete valuation ring, Bull. Amer. Math. Soc. 78 (1972), 88-92. MR 0291284 (45:378)
  • [WAR3] -, Simply presented groups, Proc. Sem. Abelian Group Theory, Univ. of Arizona lecture notes, 1972.
  • [WAR4] -, Simply presented groups, Univ. of Washington notes, January 1974.
  • [WAR5] -, A classification theorem for abelian $ p$-groups, Trans. Amer. Math. Soc. 210 (1975), 149-168. MR 0372071 (51:8288)
  • [WAR6] Classification theory of abelian groups. I. Balanced projectives, Trans. Amer. Math. Soc. 222 (1976), 33-63. MR 0422455 (54:10444)
  • [WAR7] -, Classification theory of abelian groups. II. Local theory, preprint.
  • [WAR8] -, The structure of mixed abelian groups, Abelian Group Theory (Proc. 2nd New Mexico State Univ. Conf., 1976), Lecture Notes in Math., vol. 616, Springer-Verlag, Berlin and New York, 1977, pp. 1-38. MR 0507073 (58:22342)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20K21

Retrieve articles in all journals with MSC: 20K21


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0617551-X
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society