Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-11T06:29:02.003Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generalization of the Concept of a Ring of Quotients

Published online by Cambridge University Press:  20 November 2018

John K. Luedeman
John K. Luedeman*
Affiliation:
Clemson University, Clemson, South Carolina
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Sanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 4 , December 1971 , pp. 517 - 529
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Asano, K., Über die Quotientenbildung von Schiefringen, J. Math. Soc. Japan 1 (1949), 73-78.Google Scholar
2. Bourbaki, N., Eléments de mathématique, Vol. 27, Hermann, Paris, 1961.Google Scholar
3. Eckman, B. and Schopf, A., Über Injektive Moduln, Arch. Math. 4 (1953), 75-78.Google Scholar
4. Faith, C., Lectures on infective modules and quotient rings. Springer-Verlag, New York, 1967.Google Scholar
5. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895.Google Scholar
6. Lambek, J., On Utum's ring of quotients, Canad. J. Math. 15 (1963), 363-370.Google Scholar
7. Lambek, J., Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966.Google Scholar
8. Mewborn, A. C., Generalized centralizer of modules, Duke Math. J. 35 (1968), 575-580.Google Scholar
9. Sanderson, D. F., A generalization of divisibility and injectivity in modules, Canad. Math. Bull. 8 (1965), 505-513.Google Scholar
10. Tsai, C. T., Report on injective modules, Queen's Univ., Kingston, Ontario, 1965.Google Scholar
11. Utumi, Y., On quotient rings, Osaka J. Math. 8 (1956), 1-18.Google Scholar
12. Walker, E. A. and Walker, C. P., Quotient categories of modules, Trans. Amer. Math. Soc. (to appear).Google Scholar