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Quotient Rings of a Class of Lattice-Ordered-Rings

Published online by Cambridge University Press:  20 November 2018

Stuart A. Steinberg
Stuart A. Steinberg*
Affiliation:
University of Toledo, Toledo, Ohio
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An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R

(i) (d1q)+ Λ (d2q)- = 0, and

(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.

We remark that this characterization holds even when R does not have an identity element.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 3 , June 1973 , pp. 627 - 645
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Anderson, F. W., On f-rings with the ascending chain condition, Proc. Amer. Math. Soc. 13 (1962), 715–721.Google Scholar
2. Anderson, F. W., Lattice-ordered rings of quotients, Can. J. Math. J?7 (1965), 434-448.Google Scholar
3. Bass, H., Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
4. Bigard, A., Contribution a la theorie des groupes reticules, Ph.D. Thesis, University of Paris, 1969.Google Scholar
5. Birkhoff, G. and Pierce, R. S., Lattice-ordered rings, An. Acad. Brasil. Ci. 28 (1956), 41–69.Google Scholar
6. Brungs, H. H., Generalized discrete valuation rings, Can. J. Math. 21 (1969), 1404–1408.Google Scholar
7. Conrad, P., The lattice of all convex I-subgroups of a lattice-ordered group, Czechoslovak Math. J. 15 (1965), 101–123.Google Scholar
8. Conrad, P. and Diem, J. E., The ring of polar preserving endomorphisms of an abelian latticeordered group, Illinois J. Math. 15 (1971), 222–240.Google Scholar
9. Diem, J. E., A radical for lattice-ordered rings, Pacific J. Math. 25 (1968), 71–82.Google Scholar
10. Divinsky, N., Rings and radicals (University of Toronto Press, Toronto, 1965).Google Scholar
11. Faith, C., On Kothe rings, Math. Ann. 164 (1966), 207–212.Google Scholar
12. Faith, C., Lectures on infective modules and quotient rings (Springer-Verlag, New York, 1967).Google Scholar
13. Faith, C., Rings with ascending condition on annihilators, Nagoya Math. 27 (1966), 179–191.Google Scholar
14. Faith, C. and Utumi, Y., Baer modules, Arch. Math. 15 (1964), 266–270.Google Scholar
15. Findlay, G. and Lambek, J., A generalized ring of quotients. I and II, Can. Math. Bull. 1 (1958), 77–85, 155-167.Google Scholar
16. Fuchs, L., Teilweise geordnete algebraische Strukturen (Vandenhoeck and Ruprecht in Gottingen, 1966).Google Scholar
17. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).Google Scholar
18. Goldie, A. W., Rings with maximum condition (Yale University lecture notes, New Haven, 1964).Google Scholar
19. Goldie, A. W., Torsion-free modules and rings, J. Algebra 1 (1964), 268–287.Google Scholar
20. Henricksen, M. and Isbell, J., Lattice-ordered rings and function rings, Pacific J. Math. 12 (1962), 533–565.Google Scholar
21. Jategaonkar, A. V., Left principal ideal rings, Lecture notes in Math., no. 128 (Springer- Verlag, New York, 1970).Google Scholar
22. Johnson, D. G., A structure theory for a class of lattice-ordered rings, Acta. Math. 104 (1960), 163–215.Google Scholar
23. Johnson, R. E., Extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895.Google Scholar
24. Klatt, G. B. and Levy, L. S., Pre-self-infective rings, Trans. Amer. Math. Soc. 137 (1969), 407–419.Google Scholar
25. Levy, L. S., Commutative rings whose homomorphic images are self-infective, Pacific J. Math. 18 (1966), 149–153.Google Scholar
26. Renault, G., Anneaux réduits non commutatifs, J. Math. Pures Appl. 4 (1967), 203–214.Google Scholar
27. Steinberg, S. A., An embedding theorem for commutative lattice-ordered domains, Proc. Amer. Math. Soc. 31 (1972), 409–416.Google Scholar
28. Steinberg, S. A., Finitely-valued f-modules, Pacific J. Math. 40 (1972), 723–737.Google Scholar
29. Steinberg, S. A., Lattice-ordered injective hulls, Trans. Amer. Math. Soc. 169 (1972), 365–388.Google Scholar
30. Steinberg, S. A., Rings of quotients of rings without nilpotent elements (to appear in Pacific J. Math.).Google Scholar
31. Utumi, Y., On quotient rings, Osaka J. Math. 8 (1956), 1–18.Google Scholar
32. Weinberg, E. C., Lectures on ordered groups and rings, Lecture notes (University of Illinois, Urbana, 1968).Google Scholar