Prime ideals and sheaf representation of a pseudo symmetric ring

Shin, Gooyong

doi:10.1090/S0002-9947-1973-0338058-9

Prime ideals and sheaf representation of a pseudo symmetric ring
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Trans. Amer. Math. Soc. 184 (1973), 43-60 Request permission

Abstract:

Almost symmetric rings and pseudo symmetric rings are introduced. The classes of symmetric rings, of almost symmetric rings, and of pseudo symmetric rings are in a strictly increasing order. A sheaf representation is obtained for pseudo symmetric rings, similar to the cases of symmetric rings, semiprime rings, and strongly harmonic rings. Minimal prime ideals of a pseudo symmetric ring have the same characterization, due to J. Kist, as for the commutative case. A characterization is obtained for a pseudo symmetric ring with a certain right quotient ring to have compact minimal prime ideal space, extending a result due to Mewborn.

References

Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
John Dauns and Karl Heinrich Hofmann, Representation of rings by sections, Memoirs of the American Mathematical Society, No. 83, American Mathematical Society, Providence, R.I., 1968. MR 0247487
L. Gillman, Rings with Hausdorff structure space, Fund. Math. 45 (1957), 1–16. MR 92773, DOI 10.4064/fm-45-1-1-16
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
W. D. Gwynne and J. C. Robson, Completions of non-commutative Dedekind prime rings, J. London Math. Soc. (2) 4 (1971), 346–352. MR 296109, DOI 10.1112/jlms/s2-4.2.346
M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110–130. MR 194880, DOI 10.1090/S0002-9947-1965-0194880-9
Karl Heinrich Hofmann, Representations of algebras by continuous sections, Bull. Amer. Math. Soc. 78 (1972), 291–373. MR 347915, DOI 10.1090/S0002-9904-1972-12899-4
Joseph Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. (3) 13 (1963), 31–50. MR 143837, DOI 10.1112/plms/s3-13.1.31
Kwangil Koh, On functional representations of a ring without nilpotent elements, Canad. Math. Bull. 14 (1971), 349–352. MR 369440, DOI 10.4153/CMB-1971-063-7
Kwangil Koh, Quasisimple modules and other topics in ring theory, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I), Lecture Notes in Math., Vol. 246, Springer, Berlin, 1972, pp. 323–428. MR 0335560
Kwangil Koh, On a representation of a strongly harmonic ring by sheaves, Pacific J. Math. 41 (1972), 459–468. MR 320085, DOI 10.2140/pjm.1972.41.459
J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359–368. MR 313324, DOI 10.4153/CMB-1971-065-1
A. C. Mewborn, Some conditions on commutative semiprime rings, J. Algebra 13 (1969), 422–431. MR 276209, DOI 10.1016/0021-8693(69)90084-2
R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR 0217056