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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Two-sided semisimple maximal quotient rings


Author: Vasily C. Cateforis
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
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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.


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  • [1] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [2] V. C. Cateforis, Flat regular quotient rings, Trans. Amer. Math. Soc. 138 (1969), 241-250. MR 39 #259. MR 0238899 (39:259)
  • [3] -, On regular self-injective rings, Pacific J. Math. 30 (1969), 39-45. MR 0248178 (40:1432)
  • [4] G. D. Findlay and J. Lambek, A generalized ring of quotients. I, II, Canad. Math. Bull. 1 (1958), 77-85; 155-167. MR 20 #888. MR 0094370 (20:888)
  • [5] A. W. Goldie, The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. (3) 8 (1958), 589-608. MR 21 #1988. MR 0103206 (21:1988)
  • [6] -, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201-220. MR 22 #2627. MR 0111766 (22:2627)
  • [7] J. Lambek, Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966. MR 34 #5857. MR 0206032 (34:5857)
  • [8] L. Levy, Torsion-free and divisible modules over non-integral domains, Canad. J. Math. 15 (1963), 132-151. MR 26 #155. MR 0142586 (26:155)
  • [9] -, Unique sub-direct sums of prime rings, Trans. Amer. Math. Soc. 106 (1963), 64-76. MR 26 #136. MR 0142567 (26:136)
  • [10] F. L. Sandomierski, Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112-120. MR 35 #5473. MR 0214624 (35:5473)
  • [11] Y. Utumi, On quotient rings, Osaka Math. J. 8 (1956), 1-18. MR 18, 7. MR 0078966 (18:7c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0260801-5
Keywords: Singular right ideal, Goldie dimension, maximal right quotient ring, classical right quotient ring, injective hull of a module, flat module, tensor product of modules, semisimple artinian ring, von Neumann regular ring
Article copyright: © Copyright 1970 American Mathematical Society

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