Prime and semiprime semigroup algebras of cancellative semigroups
HTML articles powered by AMS MathViewer
- by M. V. Clase PDF
- Trans. Amer. Math. Soc. 350 (1998), 1991-2007 Request permission
Abstract:
Necessary and sufficient conditions are given for a semigroup algebra of a cancellative semigroup to be prime and semiprime. These conditions were proved necessary by Okniński; our contribution is to show that they are also sufficient. The techniques used in the proof are a new variation on the $\Delta$-methods which were developed originally for group algebras.References
- Jeffrey Bergen and D. S. Passman, Delta methods in enveloping rings, J. Algebra 133 (1990), no. 2, 277–312. MR 1067408, DOI 10.1016/0021-8693(90)90271-O
- Jeffrey Bergen and D. S. Passman, Delta methods in enveloping algebras of Lie superalgebras, Trans. Amer. Math. Soc. 334 (1992), no. 1, 259–280. MR 1076611, DOI 10.1090/S0002-9947-1992-1076611-X
- Jeffrey Bergen and D. S. Passman, Delta methods in enveloping rings. II, J. Algebra 156 (1993), no. 2, 494–534. MR 1216480, DOI 10.1006/jabr.1993.1085
- Jeffrey Bergen and D. S. Passman, Delta methods in enveloping algebras of Lie superalgebras. II, J. Algebra 166 (1994), no. 3, 568–610. MR 1280592, DOI 10.1006/jabr.1994.1167
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- Ian G. Connell, On the group ring, Canadian J. Math. 15 (1963), 650–685. MR 153705, DOI 10.4153/CJM-1963-067-0
- Jan Krempa, Special elements in semigroup rings, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 1-2, 17–23 (1981) (English, with Russian summary). MR 616193
- Jan Okniński, Semigroup algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 138, Marcel Dekker, Inc., New York, 1991. MR 1083356
- Jan Okniński, Prime and semiprime semigroup rings of cancellative semigroups, Glasgow Math. J. 35 (1993), no. 1, 1–12. MR 1199933, DOI 10.1017/S0017089500009514
- D. S. Passman, Nil ideals in group rings, Michigan Math. J. 9 (1962), 375–384. MR 144930
- D. S. Passman, Radicals of twisted group rings, Proc. London Math. Soc. (3) 20 (1970), 409–437. MR 271246, DOI 10.1112/plms/s3-20.3.409
- Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
- D. S. Passman, Semiprime and prime crossed products, J. Algebra 83 (1983), no. 1, 158–178. MR 710592, DOI 10.1016/0021-8693(83)90142-4
- D. S. Passman, Infinite crossed products and group-graded rings, Trans. Amer. Math. Soc. 284 (1984), no. 2, 707–727. MR 743740, DOI 10.1090/S0002-9947-1984-0743740-2
- Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR 979094
Additional Information
- M. V. Clase
- Affiliation: 86 Herkimer St., Apt. A, Hamilton, Ontario, Canada L8P 2G7
- Address at time of publication: Andyne Computing Ltd., 1 Research Drive, Dartmouth, Nova Scotia, B2Y 4M9 Canada
- Email: michael.clase@ns.sympatico.ca
- Received by editor(s): November 17, 1995
- Received by editor(s) in revised form: August 1, 1996
- Additional Notes: This work was completed while the author held an NSERC Postdoctoral Fellowship at McMaster University, Hamilton, Canada.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 1991-2007
- MSC (1991): Primary 16S36; Secondary 16N60, 20M25
- DOI: https://doi.org/10.1090/S0002-9947-98-01922-9
- MathSciNet review: 1422893