The structure of rings with faithful nonsingular modules

Zelmanowitz, J. M.

doi:10.1090/S0002-9947-1983-0697079-3

The structure of rings with faithful nonsingular modules
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by J. M. Zelmanowitz

Trans. Amer. Math. Soc. 278 (1983), 347-359

DOI: https://doi.org/10.1090/S0002-9947-1983-0697079-3

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Abstract:

It is shown that the existence of a faithful nonsingular uniform module characterizes rings which have a full linear maximal quotient ring. New information about the structure of these rings is obtained and their maximal quotient rings are constructed in an explicit manner. More generally, rings whose maximal quotient rings are finite direct sums of full linear rings are characterized by the existence of a faithful nonsingular finite dimensional module.

References

S. A. Amitsur, Rings of quotients and Morita contexts, J. Algebra 17 (1971), 273–298. MR 414604, DOI 10.1016/0021-8693(71)90034-2
Carl Faith, Lectures on injective modules and quotient rings, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR 0227206
K. R. Goodearl, Ring theory, Pure and Applied Mathematics, No. 33, Marcel Dekker, Inc., New York-Basel, 1976. Nonsingular rings and modules. MR 0429962
R. E. Johnson, The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895. MR 45695, DOI 10.1090/S0002-9939-1951-0045695-9
R. E. Johnson, Representations of prime rings, Trans. Amer. Math. Soc. 74 (1953), 351–357. MR 53917, DOI 10.1090/S0002-9947-1953-0053917-X
R. E. Johnson, Quotient rings of rings with zero singular ideal, Pacific J. Math. 11 (1961), 1385–1392. MR 143779
R. E. Johnson and E. T. Wong, Quasi-injective modules and irreducible rings, J. London Math. Soc. 36 (1961), 260–268. MR 131445, DOI 10.1112/jlms/s1-36.1.260
Kwangil Koh and A. C. Mewborn, Prime rings with maximal annihilator and maximal complement right ideals, Proc. Amer. Math. Soc. 16 (1965), 1073–1076. MR 219573, DOI 10.1090/S0002-9939-1965-0219573-6
J. Lambek and G. O. Michler, On products of full linear rings, Publ. Math. Debrecen 24 (1977), no. 1-2, 123–127. MR 573052, DOI 10.5486/pmd.1977.24.1-2.17
W. K. Nicholson, J. F. Watters, and J. M. Zelmanowitz, On extensions of weakly primitive rings, Canadian J. Math. 32 (1980), no. 4, 937–944. MR 590656, DOI 10.4153/CJM-1980-071-6
J. Zelmanowitz, Semiprime modules with maximum conditions, J. Algebra 25 (1973), 554–574. MR 320087, DOI 10.1016/0021-8693(73)90101-4
Julius Zelmanowitz, Weakly primitive rings, Comm. Algebra 9 (1981), no. 1, 23–45. MR 599070, DOI 10.1080/00927878108822561

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