Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a question of Faith in commutative endomorphism rings

Author: John Clark
Journal: Proc. Amer. Math. Soc. 98 (1986), 196-198
MSC: Primary 13B30; Secondary 13C11, 16A65
MathSciNet review: 854017
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a commutative ring $ R$, let $ Q(R)$ denote its maximal ring of quotients and, for any ideal $ I$ of $ R$, let $ \operatorname{End} (I)$ denote the ring of $ R$-endomorphisms of $ I$. It is known that if $ Q(R)$ is a self-injective ring then $ \operatorname{End} (I)$ is commutative for each ideal $ I$ of $ R$. Carl Faith has asked if the converse holds. It does if $ R$ is either Noetherian or has no nontrivial nilpotent elements but here we produce an example to show that it does not hold in general.

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

  • [1] S. Alamelu, On commutativity of endomorphism rings of ideals, Proc. Amer. Math. Soc. 37 (1973), 29-31. MR 0311651 (47:213)
  • [2] -, Commutativity of endomorphism rings of ideals. II, Proc. Amer. Math. Soc. 55 (1976), 271-274. MR 0401731 (53:5558)
  • [3] S. H. Cox, Jr., Commutative endomorphism rings, Pacific J. Math. 45 (1975), 87-91. MR 0330143 (48:8481)
  • [4] C. Faith, Injective quotient rings of commutative rings, Module Theory (Proc. Special Session, Amer. Math. Soc., Univ. Washington, Seattle, Wash. 1977), Lecture Notes in Math., vol. 700, Springer-Verlag, Berlin, 1979, pp. 151-203. MR 550435 (81a:13014)
  • [5] -, Injective modules and injective quotient rings, Lecture Notes in Pure and Appl. Math, vol. 72, Dekker, New York, 1982. MR 643796 (83d:16023)
  • [6] J.-P. Lafon, Les formalismes fondamentaux de l'algèbre commutative, Hermann, Paris, 1974. MR 0364210 (51:465)
  • [7] J. Lambek, Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966. MR 0206032 (34:5857)
  • [8] M. Nagata, Local rings, Interscience, New York and London, 1962. MR 0155856 (27:5790)
  • [9] B. Stenström, Rings of quotients, Springer-Verlag, Berlin and New York, 1975. MR 0389953 (52:10782)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13B30, 13C11, 16A65

Retrieve articles in all journals with MSC: 13B30, 13C11, 16A65

Additional Information

Keywords: Endomorphism ring, maximal quotient ring, self-injective, trivial extension
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society