Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Prime ideals and sheaf representation of a pseudo symmetric ring


Author: Gooyong Shin
Journal: Trans. Amer. Math. Soc. 184 (1973), 43-60
MSC: Primary 16A34
DOI: https://doi.org/10.1090/S0002-9947-1973-0338058-9
MathSciNet review: 0338058
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [2] J. Dauns and K. H. Hofmann, Representation of rings by sections, Mem. Amer. Math. Soc. No. 83 (1968). MR 40 #752. MR 0247487 (40:752)
  • [3] L. Gillman, Rings with Hausdorff structure space, Fund. Math. 45 (1957), 1-16. MR 19, 1156. MR 0092773 (19:1156a)
  • [4] L. Gillman and M. Jerison, Rings of continuous functions, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [5] W. D. Gwynne and J. C. Robson, Completions of non-commutative Dedekind prime rings, J. London Math. Soc. 4 (1972), 346-352. MR 0296109 (45:5170)
  • [6] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130. MR 33 #3086. MR 0194880 (33:3086)
  • [7] K. H. Hofmann, Representations of algebras by continuous sections, Bull. Amer. Math. Soc. 78 (1972), 291-373. MR 0347915 (50:415)
  • [8] J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. (3) 13 (1963), 31-50. MR 26 #1387. MR 0143837 (26:1387)
  • [9] Kwangil Koh, On functional representations of a ring without nilpotent elements, Canad. Math. Bull. 14 (1971), 349-352. MR 0369440 (51:5673)
  • [10] -, Quasisimple modules and other topics in ring theory, Lecture Notes in Math., vol. 246, Springer-Verlag, New York, 1972. MR 0335560 (49:341)
  • [11] -, On a representation of a strongly harmonic ring by sheaves, Pacific J. Math. 41 (1972), 459-468. MR 0320085 (47:8626)
  • [12] J. Lambek, On the representations of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368. MR 0313324 (47:1879)
  • [13] A. C. Mewborn, Some conditions on commutative semiprime rings, J. Algebra 13 (1969), 422-431. MR 43 #1957. MR 0276209 (43:1957)
  • [14] R. S. Pierce, Modules over commutative regular rings, Mem. Amer. Math. Soc. No. 70 (1967). MR 36 #151. MR 0217056 (36:151)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A34

Retrieve articles in all journals with MSC: 16A34


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0338058-9
Keywords: Prime ideal, sheaf representation, pseudo symmetric ring, minimal prime ideal space
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society