Maximal Quotient Rings and S-Rings

Armendariz, E. P. ; McDonald, Gary R.

Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 835-850

Résumé

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.

Détail

DOI : 10.4153/CJM-1972-083-3

Bibliographie

[1] 1. Alin, J. S., Structure of torsion modules, Ph. D. Thesis, University of Nebraska, 1967. Google Scholar

[2] 2. Armendariz, E. P. and McDonald, G., Q-divisibility and injectivity (to appear). Google Scholar

[3] 3. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, 1956). Google Scholar

[4] 4. Cateforis, V. C., Two-sided semi-simple maximal quotient rings, Trans. Amer. Math. Soc. 149 (1970), 339–349. Google Scholar

[5] 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] 6. Jans, J. P., Some aspects of torsion, Pacific J. Math. 15 (1965), 1249–1259. Google Scholar

[7] 7. Lambek, J., Lectures on rings and modules (Blaisdell Publishing Company, Waltham, Massachusetts, 1966). Google Scholar

[8] 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] 9. Mares, E. A., Semi-perfect modules. Math. Z. 82 (1963), 347–360. Google Scholar

[10] 10. McCoy, N. H., The theory of rings (MacMillan, New York, 1964). Google Scholar

[11] 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] 12. Procesi, C. and Small, L., On a theorem of Goldie, J. Algebra 2 (1965), 80–84. Google Scholar

[13] 13. Rosenberg, A. and Zelinsky, D., Finiteness of the infective hull, Math. Z. 70 (1959), 372–380. Google Scholar

[14] 14. Sandomierski, F. L., Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120. Google Scholar

[15] 15. Small, L. W., Orders in Artinian rings, J. Algebra 4 (1966), 13–41. Google Scholar

[16] 16. Small, L. W., Orders in Artinian rings: corrections and addendum, J. Algebra 4 (1966), 505–507. Google Scholar

[17] 17. Storrer, H., Rings of quotients of perfect rings, Math. Z. 122 (1971), 151–165. Google Scholar

[18] 18. Storrer, H., Rational extensions of modules (typescript, 1970). Google Scholar

[19] 19. Walker, C. L. and Walker, E., Quotient categories and rings of quotients, Trans. Amer. Math. Soc. (to appear). Google Scholar

[20] 20. Ware, R., Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256. Google Scholar