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).

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

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 Cardinalities of prime spectra of precompletions

2021 ◽  
pp. 133-152
Author(s):  
Erica Barrett ◽  
Emil Graf ◽  
S. Loepp ◽  
Kimball Strong ◽  
Sharon Zhang
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Local Domain ◽  
Content Type ◽  
Necessary And Sufficient

Given a complete local (Noetherian) ring T T , we find necessary and sufficient conditions on T T such that there exists a local domain A A with | A | > | T | |A| > |T| and A ^ = T \widehat {A} = T , where A ^ \widehat {A} denotes the completion of A A with respect to its maximal ideal. We then find necessary and sufficient conditions on T T such that there exists a domain A A with A ^ = T \widehat {A} = T and | S p e c ( A ) | > | S p e c ( T ) | |\mathrm {Spec}(A)| > |\mathrm {Spec}(T)| . Finally, we use “partial completions” to create local rings A A with A ^ = T \widehat {A} = T such that S p e c ( A ) \mathrm {Spec}(A) has varying cardinality in different varieties.

a-Transforms of Local Rings and a Theorem on Multiplicities of Ideals

1961 ◽  
Vol 57 (1) ◽  
pp. 8-17 ◽  
Author(s):  
D. Rees
Keyword(s):  
Positive Integer ◽  
Local Ring ◽  
Maximal Ideal ◽  
Elementary Property ◽  
Local Rings ◽  
The Ideal

In two papers, (5) and (6), D. G. Northcott and the author considered the notion of the reductions of an ideal a of a Noether ring A. A reduction of a is an ideal b contained in a which satisfies ar+1 = arb for all sufficiently large r. This notion was inspired by the following elementary property of a reduction. Suppose that A is a local ring with maximal ideal m, and that a is m-primary. It is well known (Samuel (10)) that the length of the ideal an is, for large values of n equal to Pa(n) where Pa(n) is a polynomial in n whose degree d is equal to the dimension of A. If we write the coefficient of nd in Pa(n) in the form e(a)/d!, e(a) is a positive integer termed the multiplicity of a. If now b is a reduction of a, then b is also m-primary, and e(b) = e(a).

A note on the coefficients of Hilbert characteristic functions in semi-regular local rings

1963 ◽  
Vol 59 (2) ◽  
pp. 269-275 ◽  
Author(s):  
Masao Narita
Keyword(s):  
Local Ring ◽  
Characteristic Polynomial ◽  
Maximal Ideal ◽  
Negative Integer ◽  
Characteristic Functions ◽  
Local Rings ◽  
Regular Local Ring ◽  
Hilbert Coefficients ◽  
Image Position ◽  
System Of Parameters

Let Q be a semi-regular local ring of dimension d, m be its maximal ideal, and q be an m-primary ideal. Then LQ(Q/qn+1), the length of Q-module Q/qn+1, is equal to the characteristic polynomial PQ(q,n) in n for a sufficiently large value of n:where ei = ei(q), i = 0,1,2,…, d are integers uniquely determined by q, called normalized Hilbert coefficients of q according to (1). It was shown in (1) that e1(q) is a non-negative integer, and is equal to zero if and only if q is generated by a system of parameters. We shall prove, in this paper, that e2(q) is also a non-negative integer, and that this non-negativity is not necessarily true for other coefficients. We shall give an example with negative e3(q).

Rigid and Finitely V-Determined Germs of C∞-Mappings

1973 ◽  
Vol 25 (4) ◽  
pp. 727-732 ◽  
Author(s):  
Jacek Bochnak ◽  
Tzee-Char Kuo
Keyword(s):  
Maximal Ideal ◽  
Image Position ◽  
Unique Maximal Ideal ◽  
The Ideal

Let (respectively ) denote the ring of germs at 0 ∈ Rn of all C∞ functions (respectively Cμ functions) from Rn to R. For a given where is the space of all germs of C∞ mappings Rn → Rp, let J(φ) denote the ideal in generated by φ1, … , φp and the Jacobian determinantswhere LetClearly, is an ideal in and where is the (unique) maximal ideal of .

Algebraically closed commutative local rings

1979 ◽  
Vol 44 (1) ◽  
pp. 89-94 ◽  
Author(s):  
K.-P. Podewski ◽  
Joachim Reineke
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Finite System ◽  
Local Rings ◽  
Polynomial Equations ◽  
Model Companion ◽  
Local Extension ◽  
Commutative Local Ring ◽  
Finite Set ◽  
System Of Polynomial Equations

A commutative ring R with identity is called a local ring if R has only one maximal ideal. This is equivalent to saying that the sum of two nonunits is a non-unit. Therefore the theory of all commutative local rings is axiomatizible by a finite set of A2-sentences. A commutative local ring with identity is said to be an algebraically closed local ring if every finite system of polynomial equations and inequations in one or more variables with coefficients in R which has a solution in some commutative local extension of R already has a solution in R. Much work connected with algebraically closed structures of classes of rings has been done, for example by Cherlin [2], Macintyre [4] and Lipschitz and Saracino [3]. We want to show similar results for commutative local rings with identity. Our main results are the following:Theorem. The theory of commutative local rings with identity has no model-companion.The finitely generic and infinitely generic local rings are algebraically closed local rings.Theorem. There is an A3 sentence which holds for all finitely generic local rings whose negation holds in every infinitely generic local ring.

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

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].

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