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)

 
 

 

The structure of rings with faithful nonsingular modules


Author: J. M. Zelmanowitz
Journal: Trans. Amer. Math. Soc. 278 (1983), 347-359
MSC: Primary 16A48; Secondary 16A08, 16A42, 16A53, 16A64, 16A65
DOI: https://doi.org/10.1090/S0002-9947-1983-0697079-3
MathSciNet review: 697079
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the existence of a faithful nonsingular uniform module characterizes rings which have a full linear maximal quotient ring. New information about the structure of these rings is obtained and their maximal quotient rings are constructed in an explicit manner. More generally, rings whose maximal quotient rings are finite direct sums of full linear rings are characterized by the existence of a faithful nonsingular finite dimensional module.


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

  • [1] S. A. Amitsur, Rings of quotients and Morita contexts, J. Algebra 17 (1971), 273-298. MR 0414604 (54:2704)
  • [2] C. Faith, Lectures on injective modules and quotient rings, Lecture Notes in Math., vol. 49, Springer-Verlag, Berlin and New York, 1967. MR 0227206 (37:2791)
  • [3] K. R. Goodearl, Ring theory: Nonsingular rings and modules, Marcel Dekker, New York, 1976. MR 0429962 (55:2970)
  • [4] R. E. Johnson, The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895. MR 0045695 (13:618c)
  • [5] -, Representations of prime rings, Trans. Amer. Math. Soc. 74 (1953), 351-357. MR 0053917 (14:839f)
  • [6] -, Quotient rings of rings with zero singular ideal, Pacific J. Math 11 (1961), 1385-1392. MR 0143779 (26:1331)
  • [7] R. E. Johnson and E. T. Wong, Quasi-injective modules and irreducible rings, J. London Math. Soc. 36 (1961), 260-268. MR 0131445 (24:A1295)
  • [8] K. Koh and A. C. Mewborn, Prime rings with maximal annihilator and maximal complement right ideals, Proc. Amer. Math. Soc. 16 (1965), 1073-1076. MR 0219573 (36:2653)
  • [9] J. Lambek and G. O. Michler, On products of full linear rings, Publ. Math. Debrecen 24 (1977), 123-127. MR 0573052 (58:28067)
  • [10] W. K. Nicholson, J. F. Watters and J. M. Zelmanowitz, On extensions of weakly primitive rings, Canad. J. Math. 32 (1980), 937-944. MR 590656 (81m:16012)
  • [11] J. M. Zelmanowitz, Semiprime modules with maximum conditions, J. Algebra 25 (1973), 554-574. MR 0320087 (47:8628)
  • [12] -, Weakly primitive rings, Comm. Algebra 9 (1981), 23-45. MR 599070 (82j:16034)
  • [13] -, Representations of rings with faithful monoform modules.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A48, 16A08, 16A42, 16A53, 16A64, 16A65

Retrieve articles in all journals with MSC: 16A48, 16A08, 16A42, 16A53, 16A64, 16A65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0697079-3
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society