## Stability in Witt rings

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- by Thomas C. Craven PDF
- Trans. Amer. Math. Soc.
**225**(1977), 227-242 Request permission

## Abstract:

An abstract Witt ring*R*is defined to be a certain quotient of an integral group ring for a group of exponent 2. The ring

*R*has a unique maximal ideal

*M*containing 2. A variety of results are obtained concerning

*n*-stability, the condition that ${M^{n + 1}} = 2{M^n}$, especially its relationship to the ring of continuous functions from the space of minimal prime ideals of

*R*to the integers. For finite groups, a characterization of integral group rings is obtained in terms of

*n*-stability. For Witt rings of formally real fields, conditions equivalent to

*n*-stability are given in terms of the real places defined on the field.

## References

- Ron Brown,
*An approximation theorem for extended prime spots*, Canadian J. Math.**24**(1972), 167–184. MR**292802**, DOI 10.4153/CJM-1972-015-3 - Ron Brown,
*Real places and ordered fields*, Rocky Mountain J. Math.**1**(1971), no. 4, 633–636. MR**285512**, DOI 10.1216/RMJ-1971-1-4-633 - Ron Brown,
*Superpythagorean fields*, J. Algebra**42**(1976), no. 2, 483–494. MR**427286**, DOI 10.1016/0021-8693(76)90109-5 - Thomas C. Craven,
*The Boolean space of orderings of a field*, Trans. Amer. Math. Soc.**209**(1975), 225–235. MR**379448**, DOI 10.1090/S0002-9947-1975-0379448-X - Richard Elman and T. Y. Lam,
*Quadratic forms over formally real fields and pythagorean fields*, Amer. J. Math.**94**(1972), 1155–1194. MR**314878**, DOI 10.2307/2373568 - Richard Elman, Tsit Yuen Lam, and Alexander Prestel,
*On some Hasse principles over formally real fields*, Math. Z.**134**(1973), 291–301. MR**330045**, DOI 10.1007/BF01214693 - Burton W. Jones,
*The Arithmetic Theory of Quadratic Forms*, Carcus Monograph Series, no. 10, Mathematical Association of America, Buffalo, N.Y., 1950. MR**0037321**, DOI 10.5948/UPO9781614440109 - Manfred Knebusch,
*On the extension of real places*, Comment. Math. Helv.**48**(1973), 354–369. MR**337912**, DOI 10.1007/BF02566128 - Manfred Knebusch, Alex Rosenberg, and Roger Ware,
*Signatures on semilocal rings*, J. Algebra**26**(1973), 208–250. MR**327761**, DOI 10.1016/0021-8693(73)90021-5 - Manfred Knebusch, Alex Rosenberg, and Roger Ware,
*Structure of Witt rings and quotients of Abelian group rings*, Amer. J. Math.**94**(1972), 119–155. MR**296103**, DOI 10.2307/2373597 - Manfred Knebusch, Alex Rosenberg, and Roger Ware,
*Structure of Witt rings, quotients of abelian group rings, and orderings of fields*, Bull. Amer. Math. Soc.**77**(1971), 205–210. MR**271091**, DOI 10.1090/S0002-9904-1971-12683-6 - Serge Lang,
*The theory of real places*, Ann. of Math. (2)**57**(1953), 378–391. MR**53924**, DOI 10.2307/1969865 - John Milnor and Dale Husemoller,
*Symmetric bilinear forms*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR**0506372**, DOI 10.1007/978-3-642-88330-9 - A. Prestel,
*Quadratische Semi-Ordnungen und quadratische Formen*, Math. Z.**133**(1973), 319–342 (German). MR**337913**, DOI 10.1007/BF01177872 - Roger Ware,
*When are Witt rings group rings?*, Pacific J. Math.**49**(1973), 279–284. MR**332654**, DOI 10.2140/pjm.1973.49.279

## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**225**(1977), 227-242 - MSC: Primary 13K05; Secondary 13A15
- DOI: https://doi.org/10.1090/S0002-9947-1977-0424800-9
- MathSciNet review: 0424800