Geodesic
A Generalization of the Concept of a Ring of Quotients
Canadian mathematical bulletin, Tome 14 (1971) no. 4, pp. 517-529

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Luedeman, John K. A Generalization of the Concept of a Ring of Quotients. Canadian mathematical bulletin, Tome 14 (1971) no. 4, pp. 517-529. doi: 10.4153/CMB-1971-093-6
@article{10_4153_CMB_1971_093_6,
     author = {Luedeman, John K.},
     title = {A {Generalization} of the {Concept} of a {Ring} of {Quotients}},
     journal = {Canadian mathematical bulletin},
     pages = {517--529},
     year = {1971},
     volume = {14},
     number = {4},
     doi = {10.4153/CMB-1971-093-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1971-093-6/}
}
TY  - JOUR
AU  - Luedeman, John K.
TI  - A Generalization of the Concept of a Ring of Quotients
JO  - Canadian mathematical bulletin
PY  - 1971
SP  - 517
EP  - 529
VL  - 14
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1971-093-6/
DO  - 10.4153/CMB-1971-093-6
ID  - 10_4153_CMB_1971_093_6
ER  - 
%0 Journal Article
%A Luedeman, John K.
%T A Generalization of the Concept of a Ring of Quotients
%J Canadian mathematical bulletin
%D 1971
%P 517-529
%V 14
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1971-093-6/
%R 10.4153/CMB-1971-093-6
%F 10_4153_CMB_1971_093_6

[1] 1. Asano, K., Über die Quotientenbildung von Schiefringen, J. Math. Soc. Japan 1 (1949), 73-78. Google Scholar

[2] 2. Bourbaki, N., Eléments de mathématique, Vol. 27, Hermann, Paris, 1961. Google Scholar

[3] 3. Eckman, B. and Schopf, A., Über Injektive Moduln, Arch. Math. 4 (1953), 75-78. Google Scholar

[4] 4. Faith, C., Lectures on infective modules and quotient rings. Springer-Verlag, New York, 1967. Google Scholar

[5] 5. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895. Google Scholar

[6] 6. Lambek, J., On Utum's ring of quotients, Canad. J. Math. 15 (1963), 363-370. Google Scholar

[7] 7. Lambek, J., Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966. Google Scholar

[8] 8. Mewborn, A. C., Generalized centralizer of modules, Duke Math. J. 35 (1968), 575-580. Google Scholar

[9] 9. Sanderson, D. F., A generalization of divisibility and injectivity in modules, Canad. Math. Bull. 8 (1965), 505-513. Google Scholar

[10] 10. Tsai, C. T., Report on injective modules, Queen's Univ., Kingston, Ontario, 1965. Google Scholar

[11] 11. Utumi, Y., On quotient rings, Osaka J. Math. 8 (1956), 1-18. Google Scholar

[12] 12. Walker, E. A. and Walker, C. P., Quotient categories of modules, Trans. Amer. Math. Soc. (to appear). Google Scholar

Cité par Sources :