Limit distribution of the number of solutions of a system of stochastic linear homogeneous equations with a uniform coefficient matrix over a finite local ring

1998 ◽  
Vol 34 (3) ◽  
pp. 462-466 ◽  
Author(s):  
A. N. Alekseichuk
Keyword(s):  
Local Ring ◽  
Limit Distribution ◽  
Coefficient Matrix ◽  
Number Of Solutions ◽  
Finite Local Ring

The Weil Character of the Unitary Group Associated to a Finite Local Ring

2002 ◽  
Vol 54 (6) ◽  
pp. 1229-1253 ◽  
Author(s):  
Roderick Gow ◽  
Fernando Szechtman
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Weil Representation ◽  
Finite Local Ring

AbstractLetR/R be a quadratic extension of finite, commutative, local and principal rings of odd characteristic. Denote byUn(R) the unitary group of ranknassociated toR/R. The Weil representation ofUn(R) is defined and its character is explicitly computed.

Artinian Local Rings Whose Annihilating-ideal Graphs Are Star Graphs

2015 ◽  
Vol 22 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Houyi Yu ◽  
Tongsuo Wu ◽  
Weiping Gu
Keyword(s):  
Local Ring ◽  
Complete Characterization ◽  
Local Rings ◽  
Star Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Star Graphs ◽  
Finite Local Ring ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete characterization is established for a finite local ring whose annihilating-ideal graph is a star graph.

scholarly journals UNITARY GROUPS OVER LOCAL RINGS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350093 ◽  
Author(s):  
J. CRUICKSHANK ◽  
A. HERMAN ◽  
R. QUINLAN ◽  
F. SZECHTMAN
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Unitary Groups ◽  
Weil Representation ◽  
Local Rings ◽  
Hermitian Forms ◽  
Finite Local Ring

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

CONGRUENCE OF SYMMETRIC INNER PRODUCTS OVER FINITE COMMUTATIVE RINGS OF ODD CHARACTERISTIC

2017 ◽  
Vol 96 (3) ◽  
pp. 389-397
Author(s):  
SONGPON SRIWONGSA
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Finite Rank ◽  
Congruence Class ◽  
Commutative Rings ◽  
Inner Products ◽  
Finite Local Ring ◽  
Finite Commutative Ring ◽  
Finite Commutative Rings

Let $R$ be a finite commutative ring of odd characteristic and let $V$ be a free $R$-module of finite rank. We classify symmetric inner products defined on $V$ up to congruence and find the number of such symmetric inner products. Additionally, if $R$ is a finite local ring, the number of congruent symmetric inner products defined on $V$ in each congruence class is determined.

Homology, Hilbert functions, and the conservation of number

1971 ◽  
Vol 69 (1) ◽  
pp. 59-70
Author(s):  
D. G. Northcott
Keyword(s):  
Finite Number ◽  
Local Ring ◽  
Residue Field ◽  
Degree Zero ◽  
System Of Equations ◽  
Regular Local Ring ◽  
Algebraic Equations ◽  
Number Of Solutions ◽  
Homology Module ◽  
New System

1. Introduction. The principle of the Conservation of Number is concerned with the following situation. One starts with a system of algebraic equations having only a finite number of solutions and then applies a homomorphism whose domain contains the coefficients of those equations. This produces a new system. Let us suppose that the new system of equations also has only a finite number of solutions. The question then arises as to how the number of solutions before specialization compares with the number present afterwards. In a typical geometrical situation, one usually wishes to assert that the two systems have equally many solutions. However, it is easy to construct algebraic situations where the number changes,† and where the change is not to be explained away through the confluence of solutions or by their slipping off to infinity. At first sight this represents a breakdown of the conservation principle, but this principle has proved so useful in the past that one has a natural reluctance to discard it. The alternative is to attempt a reformulation and in (l) the present author gave such a reformulation for the case in which the specialization consisted in mapping a regular local ring on to its residue field. The modified theory requires that we take account of systems of equations which arise in connexion with the homology modules of a certain complex. The system associated with the homology module of degree zero is found to be the same as the one that arises in the naive theory, and usually this is the only one that makes a contribution. However, in cases where the number of solutions appears to change, the other systems become active and act in such a way that the balance is restored. For an amplification of these remarks we must refer the reader to (l). They are made here to indicate how the relevance of homological concepts first became clear in any detail. In the present paper these ideas are taken further, the principal gain being that it is no longer necessary to restrict the type of specialization to that which consists in mapping a regular local ring on to its residue field. Indeed one can use very general specializations provided that one transfers the homological requirement from the ring to the system of equations under consideration. In this way, one obtains a theory which is more general and, in some of its aspects, simpler as well.

Sign in / Sign up

Export Citation Format

Share Document