scholarly journals The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

2017 ◽  
Vol 146 (1) ◽  
pp. 1-13
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Forms ◽  
Local Rings ◽  
Residue Fields

Finite Spaces of Signatures

1989 ◽  
Vol 41 (5) ◽  
pp. 808-829 ◽  
Author(s):  
Victoria Powers
Keyword(s):  
Quadratic Forms ◽  
Local Rings ◽  
Character Group ◽  
Witt Rings ◽  
Ordered Field ◽  
Spaces Of Orderings ◽  
Ordered Fields ◽  
Skew Fields ◽  
Finite Spaces ◽  
Field Setting

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

Degree Functions in Local Rings

1961 ◽  
Vol 57 (1) ◽  
pp. 1-7 ◽  
Author(s):  
D. Rees
Keyword(s):  
Maximal Ideal ◽  
Zero Element ◽  
Transcendence Degree ◽  
Local Rings ◽  
Local Domain ◽  
Degree Function ◽  
Finite Set ◽  
Residue Fields ◽  
Image Position ◽  
The Ideal

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identityThe main purpose of this paper is to obtain a formulawhere vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).

Embedding subfields of quasi-local rings in residue fields

1962 ◽  
Author(s):  
Henry Gilbert Bray
Keyword(s):  
Local Rings ◽  
Residue Fields

scholarly journals Existentially closed models of the theory of artinian local rings

1999 ◽  
Vol 64 (2) ◽  
pp. 825-845 ◽  
Author(s):  
Hans Schoutens
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Model Companion ◽  
Closed Model ◽  
Gorenstein Rings ◽  
Existentially Closed ◽  
Finite Set ◽  
Residue Fields

AbstractThe class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms τtl. We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.

scholarly journals Reductions of ideals in local rings with finite residue fields

2010 ◽  
Vol 138 (05) ◽  
pp. 1569-1574
Author(s):  
William J. Heinzer ◽  
Louis J. Ratliff Jr. ◽  
David E. Rush
Keyword(s):  
Local Rings ◽  
Residue Fields

Quadratic forms and quadrics of space over local rings

2012 ◽  
Vol 187 (2) ◽  
pp. 177-186
Author(s):  
O. A. Starikova
Keyword(s):  
Quadratic Forms ◽  
Local Rings

scholarly journals Balanced local rings with commutative residue fields

1972 ◽  
Vol 78 (5) ◽  
pp. 771-775 ◽  
Author(s):  
Vlastimil Dlab ◽  
Claus Michael Ringel
Keyword(s):  
Local Rings ◽  
Residue Fields

scholarly journals Milnor $K$-theory of local rings with finite residue fields

2010 ◽  
Vol 19 (1) ◽  
pp. 173-191 ◽  
Author(s):  
Moritz Kerz
Keyword(s):  
Local Rings ◽  
Residue Fields ◽  
K Theory

A Note on Quadratic forms Over Arbitrary Semi-Local Rings

1975 ◽  
Vol 27 (3) ◽  
pp. 513-527
Author(s):  
K. I. Mandelberg
Keyword(s):  
Commutative Ring ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Local Rings ◽  
Quadratic Mapping ◽  
Finitely Generated ◽  
Bilinear Mapping ◽  
Natural Mapping

Let R be a commutative ring. A bilinear space (E, B) over R is â finitely generated projective R-module E together with a symmetric bilinear mapping B:E X E →R which is nondegenerate (i.e. the natural mapping E → HomR(E﹜ R) induced by B is an isomorphism). A quadratic space (E, B, ) is a bilinear space (E, B) together with a quadratic mapping ϕ:E →R such that B(x, y) = ϕ (x + y) — ϕ (x) — ϕ (y) and ϕ (rx) = r2ϕ (x) for all x, y in E and r in R. If 2 is a unit in R, then ϕ (x) = ½. B﹛x,x) and the two types of spaces are in obvious 1 — 1 correspondence.

scholarly journals Stiefel-Whitney invariants of quadratic forms over local rings

1973 ◽  
Vol 26 (2) ◽  
pp. 258-279 ◽  
Author(s):  
E.A.M Hornix
Keyword(s):  
Quadratic Forms ◽  
Local Rings
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