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Maximal Quotient Rings and S-Rings

Published online by Cambridge University Press:  20 November 2018

E. P. Armendariz  and
Gary R. McDonald
E. P. Armendariz
Affiliation:
University of Texas, Austin, Texas
Gary R. McDonald
Affiliation:
University of Texas, Austin, Texas
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Throughout, we assume all rings are associative with identity and all modules are unitary. See [7] for undefined terms and [3] for all homological concepts.

Let R be a ring, E(R) the injective envelope of RR, and H =HomR(E(R),E(R)). Then we obtain a bimodule RE(R)H. Let Q = HomH(E(R), E(R)). Q is called the maximal left quotient ring of R. Q has the property that if p, q ∈ Q, p ≠ 0, then there exists r ∈ R such that rp ≠ 0, rq ∈ R, i.e., Q is a ring of left quotients of R.

A left ideal I of R is dense if for every x,y ∈ R,x ≠ 0, there exists r ∈ R such that rx ≠ 0, ry ∈ I. An alternate description of Q is Q = {x ∈ E(RR) : (R : x) is a dense left ideal of R{, where (R : x) = {r ∈ R : rx ∈ R}.

The left singular ideal of R is Zl(R) = {r ∈ R : lR(r) is an essential left ideal of R}, where lR(r) = {x ∈ R : xr = 0}. If Zl(R) = (0), then Q is a left self-injective von Neumann regular ring [7, § 4.5]. Most of the previous work on maximal left quotient rings has been done in this case.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 835 - 850
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Alin, J. S., Structure of torsion modules, Ph. D. Thesis, University of Nebraska, 1967.Google Scholar
2. Armendariz, E. P. and McDonald, G., Q-divisibility and injectivity (to appear).Google Scholar
3. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, 1956).Google Scholar
4. Cateforis, V. C., Two-sided semi-simple maximal quotient rings, Trans. Amer. Math. Soc. 149 (1970), 339–349.Google Scholar
5. Colby, R. R. and Rutter, E. A., Jr., QF-3 rings with zero singular ideal, Pacific J. Math. 28 (1969), 303–308.Google Scholar
6. Jans, J. P., Some aspects of torsion, Pacific J. Math. 15 (1965), 1249–1259.Google Scholar
7. Lambek, J., Lectures on rings and modules (Blaisdell Publishing Company, Waltham, Massachusetts, 1966).Google Scholar
8. Luedemann, J. K., On orders in principal left ideal rings and left elemental annihilator rings. Math. Ann. 185 (1970), 309–314.Google Scholar
9. Mares, E. A., Semi-perfect modules. Math. Z. 82 (1963), 347–360.Google Scholar
10. McCoy, N. H., The theory of rings (MacMillan, New York, 1964).Google Scholar
11. Probert, G. A., Local rings whose maximal ideal is principal as a right ideal, Proc. London Math. Soc. 19 (1969), 403–420.Google Scholar
12. Procesi, C. and Small, L., On a theorem of Goldie, J. Algebra 2 (1965), 80–84.Google Scholar
13. Rosenberg, A. and Zelinsky, D., Finiteness of the infective hull, Math. Z. 70 (1959), 372–380.Google Scholar
14. Sandomierski, F. L., Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120.Google Scholar
15. Small, L. W., Orders in Artinian rings, J. Algebra 4 (1966), 13–41.Google Scholar
16. Small, L. W., Orders in Artinian rings: corrections and addendum, J. Algebra 4 (1966), 505–507.Google Scholar
17. Storrer, H., Rings of quotients of perfect rings, Math. Z. 122 (1971), 151–165.Google Scholar
18. Storrer, H., Rational extensions of modules (typescript, 1970).Google Scholar
19. Walker, C. L. and Walker, E., Quotient categories and rings of quotients, Trans. Amer. Math. Soc. (to appear).Google Scholar
20. Ware, R., Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256.Google Scholar