|
Henselian valuations and orderings of a commutative ring
Author(s):
Guangxing
Zeng
Journal:
Proc. Amer. Math. Soc.
135
(2007),
929-938.
MSC (2000):
Primary 13J30;
Secondary 13J25, 12J15, 12D15
Posted:
September 18, 2006
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
The purpose of this paper is to investigate the interplay between henselian valuations and orderings (or semiorderings) of a ring. As a main result, it is proved that for a henselian valuation  on a ring  , the following statements are equivalent: (1)  is compatible with every semiordering of  ; (2)  is compatible with every ordering of  ; (3) Every real prime ideal of  is contained in the core of  .
References:
-
- 1.
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969. MR 0242802 (39:4129)
- 2.
- N. Bourbaki, Algébre commutative, Hermann, Paris, 1964. MR 0194450 (33:2660)
- 3.
- Z. Dai, Real places and real valuations on a commutative ring, Acta Math. Sinica, Special Issue, 10(1994), 14-25. MR 1268256 (95b:13003)
- 4.
- O. Endler, Valuation Theory, Springer-Verlag, New York, 1972. MR 0357379 (50:9847)
- 5.
- E. Enochs, Totally integrally closed rings, Proc. Amer. Math. Soc. 19(1968), 701-706. MR 0224600 (37:199)
- 6.
- M. Hochster, Totally integrally closed rings and extremal spaces, Pacific J. Math. 32(1970), 767-779. MR 0257064 (41:1718)
- 7.
- T. Y. Lam, The Theory of Ordered Fields, Lecture Notes in Pure and Appl. Math. 55, M. Dekker, New York, 1980. MR 0584611 (82e:12033)
- 8.
- T. Y. Lam, An introduction to real algebras, Rocky Mountain J. Math. 14(1984), 767-814. MR 0773114 (86g:12013)
- 9.
- M. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. 20(1969), 193-198. MR 0233813 (38:2134)
- 10.
- M. Marshall, Orderings and real places on a commutative ring, J. Algebra 140(1991), 484-501. MR 1120436 (92g:13030)
- 11.
- A. Prestel, Lecture on formally real fields, Lecture Notes in Math. 1093, Springer-Verlag, Berlin, New York, 1984. MR 0769847 (86h:12013)
- 12.
- N. Sankaran and K. Varadarajan, Formally real rings and their real closures, Acta Math. Hungarica 70(1996), 101-120. MR 1361464 (96h:13022)
- 13.
- G. Zeng, On semireal closed rings, Comm. in Algebra 29(2001), 2171-2183. MR 1837969 (2002f:13050)
- 14.
- G. Zeng, On real closed rings of higher level, J. Algebra 290(2005), 250-265. MR 2154992
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
13J30,
13J25, 12J15, 12D15
Retrieve articles in all Journals with MSC
(2000):
13J30,
13J25, 12J15, 12D15
Additional Information:
Guangxing
Zeng
Affiliation:
Department of Mathematics, Nanchang University, Jiangxi Province, Nanchang 330047, People's Republic of China
Email:
zenggx@ncu.edu.cn
DOI:
10.1090/S0002-9939-06-08726-0
PII:
S 0002-9939(06)08726-0
Keywords:
Commutative ring,
henselian valuation,
ordering,
real prime ideal
Received by editor(s):
October 23, 2005
Posted:
September 18, 2006
Additional Notes:
This work was partially supported by a National Key Basic Research Project of China (Grant No. 2004CB318003).
Communicated by:
Bernd Ulrich
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org