Two-sided semisimple maximal quotient rings
HTML articles powered by AMS MathViewer
- by Vasily C. Cateforis PDF
- Trans. Amer. Math. Soc. 149 (1970), 339-349 Request permission
Abstract:
Let R be an associative ring with singular right ideal zero and finite right Goldie dimension; F. L. Sandomierski has shown that the (R. E. Johnson) maximal right quotient ring Q of R is then semisimple (artinian). In this paper necessary and sufficient conditions are sought that Q be also a left (necessarily the maximal) quotient ring of R. Flatness of Q as a right R-module is shown to be such a condition. The condition that R have singular left ideal zero and finite left Goldie dimension, though necessary, is shown to be not sufficient in general. Conditions of two-sidedness of Q are also obtained in terms of the homogeneous components (simple subrings) of Q and the subrings of R, they induce.References
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- Vasily C. Cateforis, Flat regular quotient rings, Trans. Amer. Math. Soc. 138 (1969), 241–249. MR 238899, DOI 10.1090/S0002-9947-1969-0238899-1
- Vasily C. Cateforis, On regular self-injective rings, Pacific J. Math. 30 (1969), 39–45. MR 248178
- G. D. Findlay and J. Lambek, A generalized ring of quotients. I, II, Canad. Math. Bull. 1 (1958), 77–85, 155–167. MR 94370, DOI 10.4153/CMB-1958-009-3
- A. W. Goldie, The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. (3) 8 (1958), 589–608. MR 103206, DOI 10.1112/plms/s3-8.4.589
- A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220. MR 111766, DOI 10.1112/plms/s3-10.1.201
- Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
- Lawrence Levy, Torsion-free and divisible modules over non-integral-domains, Canadian J. Math. 15 (1963), 132–151. MR 142586, DOI 10.4153/CJM-1963-016-1
- Lawrence Levy, Unique subdirect sums of prime rings, Trans. Amer. Math. Soc. 106 (1963), 64–76. MR 142567, DOI 10.1090/S0002-9947-1963-0142567-9
- Francis L. Sandomierski, Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120. MR 214624, DOI 10.1090/S0002-9947-1967-0214624-3
- Yuzo Utumi, On quotient rings, Osaka Math. J. 8 (1956), 1–18. MR 78966
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 149 (1970), 339-349
- MSC: Primary 16.80
- DOI: https://doi.org/10.1090/S0002-9947-1970-0260801-5
- MathSciNet review: 0260801