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Maximal Quotient Rings of Endomorphism Rings of E(RR)-Torsionfree Generators

Published online by Cambridge University Press:  20 November 2018

Tatsuo Izawa
Tatsuo Izawa*
Affiliation:
Shizuoka University, Ohya, Shizuoka, Japan
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Let R be a ring with identity and let H = End (E(RR)) and Q = Dou(E(RR)) = End(HE(RR)). Then Lambek [11] showed that Q is always isomorphic to Qm(R), the maximal right quotient ring of R. And Johnson [10] and Wong-Johnson [26] proved that Qm(R) is regular and right self-injective if and only if R is right non-singular, and then H is isomorphic to Qm(R), too. Moreover, Sandomierski [18] showed that Qm(R) is semi-simple Artinian if and only if R is right finite dimensional and right non-singular. And it is well known that Qm(R) is a quasi-Frobenius ring if and only if E(RR) is a rational extension of RR and the ACC holds on right annihilators of subsets of E(RR).

The purpose of this paper is to give some module-theoretic generalizations of these results. Let PR be an E(PR)-torsionless generator, and let S = End(PR), H = End(E(PR)) and Q = Dou(E(PR)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 3 , 01 June 1981 , pp. 585 - 605
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Anderson, F., Endomorphism rings of projective modules, Math. Z. 111 (1969), 322–332.Google Scholar
2. Cunningham, R. S., Rutter, E. A., Jr. and Turnidge, D. R., Rings of quotients of endomorphism rings of projective modules, Pacific J. Math. 41 (1972), 647–668.Google Scholar
3. Faith, C., Lectures on infective modules and quotient rings (Springer, Berlin-Heidelberg- New York, 1964).Google Scholar
4. Findlay, G. D. and Lambek, J., A generalized rings of quotients, I, II, Can. Math. Bull. 2 (1958), 77–85, 155-167.Google Scholar
5. Gabriel, P., Des catégories abéliannes, Bull. Soc. Math. France 90 (1962), 323–448.Google Scholar
6. Golan, J., Localization of noncommutative rings (Marcel Dekker Inc., 1975).Google Scholar
7. Goldman, O., Rings and modules of quotients, J. Algebra 13 (1969), 10–47.Google Scholar
8. Hirata, K., Some types of separable extensions of rings, Nagoya Math. J. 33 (1968), 107–115.Google Scholar
9. Izawa, T., Torsion theories and torsionless generators, Rep Fac. Sci. Shizuoka Univ. 11 (1976), 1–8.Google Scholar
10. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895.Google Scholar
11. Lambek, J., On Utumïs ring of quotients, Can. J. Math. 15 (1963), 363–370.Google Scholar
12. Lambek, J., Lectures on rings and modules (Blaisdell Waltham, Mass., 1966).Google Scholar
13. Lambek, J., Torsion theories, additive semantics, and rings of quotients (Springer, Berlin- Heidelberg-New York, 1971).Google Scholar
14. Mares, E. A., Semi-perfect modules, Math. Z. 82 (1963), 347–360.Google Scholar
15. Masaike, K., Quasi-Frobenius maximal quotient rings, Sci. Rep. Tokyo Kyoiku D., Sec. A. 11 (1971), 1–5.Google Scholar
16. Mueller, B. J., On semi-perfect rings, 111. J. Math. 14 (1970), 464–467.Google Scholar
17. Osofsky, B. L., Endomorphism rings of quasi-infective modules, Can. J. Math. 20 (1968), 895–903.Google Scholar
18. Sandomierski, F. L., Semi-simple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120.Google Scholar
19. Sandomierski, F. L., Some examples of right self-infective rings which are not left self-infective, Proc. Amer. Math. Soc. 26 (1970), 244–245.Google Scholar
20. Shock, R. C., The ring of endomorphism s of a finite dimensional module, Israel J. Math. 11 (1972), 309–314.Google Scholar
21. Stenstrôm, B., Rings and modules of quotients (Springer, Berlin-Heidelberg-New York, 1971).Google Scholar
22. Stenstrôm, B., Rings of quotients (Springer, Berlin-Heidelberg-New York, 1975).Google Scholar
23. Turnidge, D. R., Torsion theories and rings of quotients of Morita equivalent rings, Pacific J. Math. 87 (1971), 225–234.Google Scholar
24. Utumi, Y., On quotient rings, Osaka Math. J. 8 (1956), 1–18.Google Scholar
25. Utumi, Y., On continuous rings and self-infective rings, Trans. Amer. Math. Soc. 118 (1965), 158–173.Google Scholar
26. Wong, E. T. and Johnson, R. E., Self-infective rings, Can. Math. Bull. 2 (1959), 167–173.Google Scholar
27. Zelmanowitz, J. M., Endomorphism rings of torsionless modules, J. Algebra 5 (1967), 325–341.Google Scholar