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)

 
 

 

Hereditary $ QI$-rings


Author: Ann K. Boyle
Journal: Trans. Amer. Math. Soc. 192 (1974), 115-120
MSC: Primary 16A52
DOI: https://doi.org/10.1090/S0002-9947-1974-0338075-X
MathSciNet review: 0338075
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider in this paper rings in which every quasi-injective right R-module is injective. These rings will be referred to as right QI-rings. For a hereditary ring, this is equivalent to the condition that R be noetherian and a right V-ring. We also consider rings in which proper cyclic right R-modules are injective. These are right QI-rings which are either semisimple or right hereditary, right Ore domains in which indecomposable injective right R-modules are either simple or isomorphic to the injective hull of $ {R_R}$.


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

  • [1] R. Baer, Abelian groups that are direct summands of every containing abelian group, Bull. Amer. Math. Soc. 46 (1940), 800-806. MR 2, 126. MR 0002886 (2:126i)
  • [2] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [3] J. Cozzens, Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75-79. MR 41 #3531. MR 0258886 (41:3531)
  • [4] C. Faith, Algebra: Rings, modules and categories (unpublished).
  • [5] C. Faith and E.A. Walker, Direct sum representations of injective modules, J. Algebra 5 (1967), 203-221. MR 34 #7575. MR 0207760 (34:7575)
  • [6] E. Gentile, On rings with one-sided field of quotients, Proc. Amer. Math. Soc. 11 (1960), 380-384. MR 23 #A187. MR 0122855 (23:A187)
  • [7] A. W. Goldie, The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. (3) 8 (1958), 589-608. MR 21 #1988. MR 0103206 (21:1988)
  • [8] R. E. Johnson and E.T. Wong, Quasi-injective modules and irreducible rings, J. London Math. Soc. 36 (1961), 260-268. MR 24 #A1295. MR 0131445 (24:A1295)
  • [9] R. P. Kurshan, Rings whose cyclic modules have finitely generated socle, J. Algebra 15 (1963), 376-386. MR 0260780 (41:5403)
  • [10] L. Levy, Torsion-free and divisible modules over nonintegral domains, Canad. J. Math. 15 (1963), 132-151. MR 26 #155. MR 0142586 (26:155)
  • [11] B. Mitchell, Theory of categories, Pure and Appl. Math., vol. 17, Academic Press, New York, 1965. MR 34 #2647. MR 0202787 (34:2647)
  • [12] F. Sandomierski, Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112-120. MR 35 #5473. MR 0214624 (35:5473)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A52

Retrieve articles in all journals with MSC: 16A52


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0338075-X
Keywords: Right V-ring, simple module, hereditary ring, QI-ring, quasi-injective module, injective module, torsion module, proper cyclic, PCI-ring
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society