Gorenstein Witt Rings

1988 ◽  
Vol 40 (5) ◽  
pp. 1186-1202 ◽  
Author(s):  
Robert W. Fitzgerald
Keyword(s):  
Group Ring ◽  
Quotient Ring ◽  
Injective Resolution ◽  
Gorenstein Rings ◽  
Witt Ring ◽  
Witt Rings ◽  
Local Type ◽  
Elementary Type ◽  
Ring Extensions

Throughout R is a noetherian Witt ring. The basic example is the Witt ring WF of a field F of characteristic not 2 and finite. We study the structure of (noetherian) Witt rings which are also Gorenstein rings (i.e., have a finite injective resolution). The underlying motivation is the elementary type conjecture. The Gorenstein Witt rings of elementary type are group ring extensions of Witt rings of local type. We thus wish to compare the two classes of Witt rings: Gorenstein and group ring over local type. We show the two classes enjoy many of the same properties and are, in several cases, equal. However we cannot decide if the two classes are always equal.In the first section we consider formally real Witt rings R (equivalently, dim R = 1). Here the total quotient ring of R is R-injective if and only if R is reduced. Further, R is Gorenstein if and only if R is a group ring over Z. This result appears to be somewhat deep.

Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements

1993 ◽  
Vol 45 (6) ◽  
pp. 1184-1199 ◽  
Author(s):  
Craig M. Cordes
Keyword(s):  
Group Ring ◽  
Positive Characteristic ◽  
Finitely Generated ◽  
Binary Forms ◽  
Witt Ring ◽  
Witt Rings ◽  
Elementary Type ◽  
Characteristic 2

AbstractAn abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.

Decomposition of Witt Rings and Galois Groups

1995 ◽  
Vol 47 (6) ◽  
pp. 1274-1289 ◽  
Author(s):  
Ján Mináč ◽  
Tara L. Smith
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Groups ◽  
Complete List ◽  
Direct Products ◽  
Free Products ◽  
Basic Part ◽  
Witt Ring ◽  
Semidirect Products ◽  
Witt Rings

AbstractTo each field F of characteristic not 2, one can associate a certain Galois group 𝒢F, the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paper we show that direct products of Witt rings correspond to free products of these Galois groups (in the appropriate category), group ring construction of Witt rings corresponds to semidirect products of W-groups, and the basic part of W(F) is related to the center of 𝒢F. In an appendix we provide a complete list of Witt rings and corresponding w-groups for fields F with |Ḟ/Ḟ2| ≤ 16.

Decomposition of Witt Rings

1982 ◽  
Vol 34 (6) ◽  
pp. 1276-1302 ◽  
Author(s):  
Andrew B. Carson ◽  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Classification Problem ◽  
Bilinear Forms ◽  
Interesting Problem ◽  
Witt Ring ◽  
Witt Rings ◽  
Finite Case ◽  
Characteristic 2 ◽  
Definition Of

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

Signatures and Semi Signatures of Abstract Witt Rings and Witt Rings of Semilocal Rings

1978 ◽  
Vol 30 (4) ◽  
pp. 872-895 ◽  
Author(s):  
Jerrold L. Kleinstein ◽  
Alex Rosenberg
Keyword(s):  
Residue Class ◽  
Point Of View ◽  
Prime Ideals ◽  
Bilinear Forms ◽  
Semilocal Ring ◽  
Witt Ring ◽  
Witt Rings ◽  
Full Knowledge ◽  
Carry Over

This paper originated in an attempt to carry over the results of [3] from the case of a field of characteristic different from two to that of semilocal rings. To carry this out, we reverse the point of view of [3] and do assume a full knowledge of the theory of Witt rings of classes of nondegenerate symmetric bilinear forms over semilocal rings as given, for example, in [10; 11]. It turns out that the rings WT of [3] are just the residue class rings of W(C), the Witt ring of a semilocal ring C, modulo certain intersections of prime ideals.

Graded witt rings of elementary type

1985 ◽  
Vol 272 (2) ◽  
pp. 267-280 ◽  
Author(s):  
J�n Kr. Arason ◽  
Richard Elman ◽  
Bill Jacob
Keyword(s):  
Witt Rings ◽  
Elementary Type

Isomorphisms and Automorphisms of Witt Rings

1988 ◽  
Vol 31 (2) ◽  
pp. 250-256 ◽  
Author(s):  
David Leep ◽  
Murray Marshall
Keyword(s):  
Quadratic Forms ◽  
Multiplicative Group ◽  
Witt Ring ◽  
Witt Rings ◽  
Ring Isomorphism ◽  
Characteristic 2 ◽  
The Relationship

AbstractFor a field F, char(F) ≠ 2, let WF denote the Witt ring of quadratic forms of F and let denote the multiplicative group of 1-dimensional forms It follows from a construction of D. K. Harrison that if E, F are fields (both of characteristic ≠ 2) and ρ.WE → WF is a ring isomorphism, then there exists a ring isomorphism which “preserves dimension” in the sense that In this paper, the relationship between ρ and is clarified.

Hulls of Ring Extensions

2010 ◽  
Vol 53 (4) ◽  
pp. 587-601 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jae Keol Park ◽  
S. Tariq Rizvi
Keyword(s):  
Group Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Affirmative Answer ◽  
Infinite Matrix ◽  
Matrix Rings ◽  
Morita Equivalent ◽  
Semiprime Rings ◽  
Ring Extensions ◽  
The Right

AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

scholarly journals Semiorderings and Witt rings

2003 ◽  
Vol 67 (2) ◽  
pp. 329-341 ◽  
Author(s):  
Thomas C. Craven ◽  
Tara L. Smith
Keyword(s):  
Covering Number ◽  
Witt Ring ◽  
Witt Rings ◽  
Pythagorean Field

For a pythagorean field F with semiordering Q and associated preordering T, it is shown that the Witt ring WT (F) is isomorphic to the Witt ring W (K) whre K is a closure of F with respect to Q. For an arbitrary preordering T, it is shown how the covering number of T relates to the construction of WT (F).

scholarly journals Twisted group rings which are semi-prime Goldie rings

1975 ◽  
Vol 16 (1) ◽  
pp. 1-11 ◽  
Author(s):  
A. Reid
Keyword(s):  
Group Ring ◽  
Quotient Ring ◽  
Factor Group ◽  
Polynomial Rings ◽  
Group Rings ◽  
Twisted Group ◽  
Twisted Group Ring ◽  
Quotient Rings

In this paper we examine when a twisted group ring,Rγ(G), has a semi-simple, artinian quotient ring. In §1 we assemble results and definitions concerning quotient rings, Ore sets and Goldie rings and then, in §2, we defineRγ(G). We prove a useful theorem for constructing a twisted group ring of a factor group and establish an analogue of a theorem of Passman. Twisted polynomial rings are discussed in §3 and I am indebted to the referee for informing me of the existence of [4]. These are used as a tool in proving results in §4.

scholarly journals Automorphisms of Witt rings of local type

2015 ◽  
Vol 14 (3) ◽  
pp. 109-119
Author(s):  
Marcin Ryszard Stepien ◽  
Keyword(s):  
Witt Rings ◽  
Local Type
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