Hostname: page-component-6bf8c574d5-5ws7s Total loading time: 0 Render date: 2025-03-09T02:00:53.508Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of finite commutative rings constructed from Witt rings

Published online by Cambridge University Press:  17 April 2009

Thomas Craven  and
Monika Vo
Thomas Craven
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822, United States of America, e-mail: tom@math.hawaii.edu
Monika Vo
Affiliation:
Department of Mathematics and Sciences, Saint Leo University, Saint Leo, FL 33574, United States of America, e-mail: monika.vo@saintleo.edu
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Motivated by constructions of Witt rings in the algebraic theory of quadratic forms, the authors construct new classes of finite commutative rings and explore some of their properties. These rings are constructed as quotient rings of a special class of integral group rings for which the group is an elementary 2-group. The new constructions are compared to other rings in the literature.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 1 , February 2006 , pp. 47 - 64
Copyright
Copyright © Australian Mathematical Society 2006

References

Referenes

[1]Andradas, C., Bröcker, L. and Ruiz, J., Constructible sets in real geometry (Springer-Verlag, Berlin, 1966).Google Scholar
[2]Craven, T., ‘Stability in Witt rings’, Trans. Amer. Math. Soc. 225 (1977), 227–242.CrossRefGoogle Scholar
[3]Craven, T., ‘Characterizing reduced Witt rings of fields’, J. Algebra 53 (1978), 68–77.CrossRefGoogle Scholar
[4]Craven, T., ‘Fields maximal with respect to a set of orderings’, J. Algebra 115 (1988), 200–218.CrossRefGoogle Scholar
[5]Elman, R., Lam, T.Y. and Wadsworth, A.R., ‘Pfister ideals in Witt rings‘, Math. Ann. 245 (1979), 219–245.CrossRefGoogle Scholar
[6]Fitzgerald, R. and Yucas, J., ‘Combinatorial techniques and abstract Witt rings, II’, Rocky Mountain J. Math. 19 (1989), 687–708.CrossRefGoogle Scholar
[7]Kleinstein, J. and Rosenberg, A., ‘Signatures and semisignatures of abstract Witt rings and Witt rings of semilocal rings’, Canadian J. Math. 30 (1978), 872–895.CrossRefGoogle Scholar
[8]Knebusch, M., Rosenberg, A. and Ware, R., ‘Structure of Witt rings and quotients of abelian group rings’, American J. Math. 94 (1972), 119–155.CrossRefGoogle Scholar
[9]Knebusch, M., Rosenberg, A. and Ware, R., ‘Signatures on semilocal rings’, J. Algebra 26 (1973), 208–250.CrossRefGoogle Scholar
[10]Lam, T.Y., The algebraic theory of quadratic forms, Mathematics Lecture Note Series, (Revised second printing) (Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980).Google Scholar
[11]Lam, T.Y., Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics 52 (Amer. Math. Soc., Providence, RI, 1983).CrossRefGoogle Scholar
[12]MacDonald, B.R., Finite rings with identity, Pure and Applied Mathematics 28 (Marcel Dekker, New York, 1974).Google Scholar
[13]Marshall, M., Abstract Witt rings, Queen's Papers in Pure and Appl. Math. 57 (Queen's University, Kingston, Ontario, 1980).Google Scholar
[14]Marshall, M., ‘The Witt ring of a space of orderings’, Trans. Amer. Math. Soc. 258 (1980), 505–521.Google Scholar
[15]Vo, M., New classes of finite commutative rings, (Ph.D. thesis) (University of Hawaii, Honolulu, HI, 2003).Google Scholar