Polynomials inducing the zero function on chain rings

2018 ◽  
Vol 17 (08) ◽  
pp. 1850160 ◽  
Author(s):  
Mark W. Rogers ◽  
Cameron Wickham
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Minimal Set ◽  
Principal Ideal Ring ◽  
Ideal Formula ◽  
Zero Function ◽  
The Ideal

We provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that map the maximal ideal [Formula: see text] into one of its powers [Formula: see text], where [Formula: see text] is a discrete valuation ring with a finite residue field. We use this to provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that send [Formula: see text] to zero, where [Formula: see text] is a finite commutative local principal ideal ring.

Discriminant as a product of local discriminants

2017 ◽  
Vol 16 (10) ◽  
pp. 1750198 ◽  
Author(s):  
Anuj Jakhar ◽  
Bablesh Jhorar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Maximal Ideal ◽  
Valuation Ring ◽  
Approximation Theorem ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Weak Approximation ◽  
Separable Extension ◽  
Ideal Formula ◽  
Valuation Rings ◽  
Basic Equality

Let [Formula: see text] be a discrete valuation ring with maximal ideal [Formula: see text] and [Formula: see text] be the integral closure of [Formula: see text] in a finite separable extension [Formula: see text] of [Formula: see text]. For a maximal ideal [Formula: see text] of [Formula: see text], let [Formula: see text] denote respectively the valuation rings of the completions of [Formula: see text] with respect to [Formula: see text]. The discriminant satisfies a basic equality which says that [Formula: see text]. In this paper, we extend the above equality on replacing [Formula: see text] by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.

On projective intersection graph of ideals of commutative rings

2019 ◽  
pp. 2150017
Author(s):  
V. Ramanathan
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Artinian Ring ◽  
Intersection Graph ◽  
Principal Ideal Ring ◽  
Ideal Formula ◽  
Ideal Ring ◽  
Projective Graph ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

Special Principal Ideal Rings and Absolute Subretracts

1991 ◽  
Vol 34 (3) ◽  
pp. 364-367 ◽  
Author(s):  
Eric Jespers
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Identity Mapping ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Theoretic Version

AbstractA ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism f from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs.Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this paper (finite) special principal ideal rings that are absolute subretracts are studied.

Noetherian domains in which every ideal is isomorphic to a trace ideal

2019 ◽  
Vol 19 (10) ◽  
pp. 2050200
Author(s):  
A. Mimouni
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Large Family ◽  
One Dimensional ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Trace Ideal

This paper seeks an answer to the following question: Let [Formula: see text] be a Noetherian ring with [Formula: see text]. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain [Formula: see text] with [Formula: see text], every ideal is isomorphic to a trace ideal if and only if either [Formula: see text] is a DVR or [Formula: see text] is one-dimensional divisorial domain, [Formula: see text] is a principal ideal of [Formula: see text] and [Formula: see text] posses the property that every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. Next, we globalize our result by showing that a Noetherian domain [Formula: see text] with [Formula: see text] has every ideal isomorphic to a trace ideal if and only if either [Formula: see text] is a PID or [Formula: see text] is one-dimensional divisorial domain, every invertible ideal of [Formula: see text] is principal and for every non-invertible maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] is a principal ideal of [Formula: see text] and every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

scholarly journals On the lifting of the Dade group

2019 ◽  
Vol 22 (3) ◽  
pp. 441-451
Author(s):  
Caroline Lassueur ◽  
Jacques Thévenaz
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Group Homomorphism ◽  
Complete Discrete Valuation Ring ◽  
Dade Group

Abstract For the group of endo-permutation modules of a finite p-group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic p. We prove that this reduction map always has a section which is a group homomorphism.

Extending Endo-monomial Modules

2013 ◽  
Vol 20 (01) ◽  
pp. 169-172
Author(s):  
Ziqun Lu ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Characteristic P ◽  
Complete Discrete Valuation Ring ◽  
A Finite Group

Let G be a finite group with a normal Sylow p-subgroup P. Let [Formula: see text] be a complete discrete valuation ring with residue field F of characteristic p. Let M be an indecomposable endo-monomial [Formula: see text]-module. In this paper we prove that M extends to an [Formula: see text]-module if and only if M is G-stable. A similar and well-known version for endo-permutation modules is due to Dade.

scholarly journals Amount algebras

2021 ◽  
pp. 2250102
Author(s):  
Peyman Nasehpour
Keyword(s):  
Positive Integer ◽  
Valuation Ring ◽  
Prüfer Domain ◽  
Radical Ideal ◽  
Ideal Formula ◽  
Torsion Free ◽  
The Ideal

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson–Badawi [Formula: see text] conjecture, we prove that under some conditions, the formula [Formula: see text] holds for some amount [Formula: see text]-algebras [Formula: see text] and some ideals [Formula: see text] of [Formula: see text], where [Formula: see text] is the smallest positive integer [Formula: see text] that the ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing. A corollary to the mentioned formula is that if, for example, [Formula: see text] is a Prüfer domain or a torsion-free valuation ring and [Formula: see text] is a radical ideal of [Formula: see text], then [Formula: see text].

scholarly journals The First Monsky-Washnitzer Cohomology Group

1972 ◽  
Vol 48 ◽  
pp. 99-128
Author(s):  
David Meredith
Keyword(s):  
Finite Type ◽  
Characteristic Zero ◽  
Cohomology Group ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Perfect Field ◽  
Quotient Field ◽  
Complete Discrete Valuation Ring

Throughout this paper, k is a perfect field of characteristic p > 0, R is a complete discrete valuation ring with residue field k and quotient field of characteristic zero, and Z is a connected smooth prescheme of finite type over k.

Witts Theorem for Quadratic Forms Over Non-Dyadic Discrete Valuation Rings

1977 ◽  
Vol 29 (5) ◽  
pp. 928-936
Author(s):  
David Mordecai Cohen
Keyword(s):  
Bilinear Form ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Symmetric Bilinear Form ◽  
Finitely Generated ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Group Of Isometries

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).

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