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.

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).

A transfer theorem for Henselian valued and ordered fields

1993 ◽  
Vol 58 (3) ◽  
pp. 915-930 ◽  
Author(s):  
Rafel Farré
Keyword(s):  
Residue Field ◽  
Valued Fields ◽  
Transfer Theorem ◽  
Ordered Field ◽  
Existentially Closed ◽  
Henselian Valued Fields ◽  
Ordered Fields ◽  
Elementary Embeddings ◽  
Residue Fields ◽  
Natural Way

AbstractIn well-known papers ([A-K1], [A-K2], and [E]) J. Ax, S. Kochen, and J. Ershov prove a transfer theorem for henselian valued fields. Here we prove an analogue for henselian valued and ordered fields. The orders for which this result apply are the usual orders and also the higher level orders introduced by E. Becker in [Bl] and [B2]. With certain restrictions, two henselian valued and ordered fields are elementarily equivalent if and only if their value groups (with a little bit more structure) and their residually ordered residue fields (a henselian valued and ordered field induces in a natural way an order in its residue field) are elementarily equivalent. Similar results are proved for elementary embeddings and ∀-extensions (extensions where the structure is existentially closed).

A minimal ring extension of a large finite local prime ring is probably ramified

2019 ◽  
Vol 19 (01) ◽  
pp. 2050015 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Isomorphism Class ◽  
Residue Field ◽  
Galois Ring ◽  
Local Rings ◽  
Ring Extension ◽  
Algebra Isomorphism ◽  
Isomorphism Classes ◽  
Finite Set ◽  
Minimal Ring Extension ◽  
Historical Remarks

Given any minimal ring extension [Formula: see text] of finite fields, several families of examples are constructed of a finite local (commutative unital) ring [Formula: see text] which is not a field, with a (necessarily finite) inert (minimal ring) extension [Formula: see text] (so that [Formula: see text] is a separable [Formula: see text]-algebra), such that [Formula: see text] is not a Galois extension and the residue field of [Formula: see text] (respectively, [Formula: see text]) is [Formula: see text] (respectively, [Formula: see text]). These results refute an assertion of G. Ganske and McDonald stating that if [Formula: see text] are finite local rings such that [Formula: see text] is a separable [Formula: see text]-algebra, then [Formula: see text] is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let [Formula: see text] be a finite special principal ideal ring (SPIR), but not a field, such that [Formula: see text] has index of nilpotency [Formula: see text] ([Formula: see text]). Impose the uniform distribution on the (finite) set of ([Formula: see text]-algebra) isomorphism classes of the minimal ring extensions of [Formula: see text]. If [Formula: see text] (for instance, if [Formula: see text]), the probability that a random isomorphism class consists of ramified extensions of [Formula: see text] is at least [Formula: see text]; if [Formula: see text] (for instance, if [Formula: see text] for some odd prime [Formula: see text]), the corresponding probability is at least [Formula: see text]. Additional applications, examples and historical remarks are given.

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.

scholarly journals Characterizations of regular local rings via syzygy modules of the residue field

2018 ◽  
Vol 10 (3) ◽  
pp. 327-337
Author(s):  
Dipankar Ghosh ◽  
Anjan Gupta ◽  
Tony J. Puthenpurakal
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Syzygy Modules ◽  
Regular Local Rings

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

1992 ◽  
Vol 111 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Yinghwa Wu
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Primary Ideal ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Higher Dimensional ◽  
Reduction Number ◽  
Minimal Reduction ◽  
System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

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].

The Poincaré Series Of Stretched Cohen-Macaulay Rings

1980 ◽  
Vol 32 (5) ◽  
pp. 1261-1265 ◽  
Author(s):  
Judith D. Sally
Keyword(s):  
Local Ring ◽  
Poincaré Series ◽  
Local Rings ◽  
Embedding Dimension ◽  
Loop Spaces ◽  
Gorenstein Rings ◽  
Poincare Series ◽  
Image Position

There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré serieswhere k = R/m, has been settled. (For an example of a local ring with non-rational Poincaré series see the recent paper by D. Anick, “Construction of loop spaces and local rings whose Poincaré—Betti series are nonrational”, C. R. Acad. Sc. Paris 290 (1980), 729-732.) In this note, we compute the Poincaré series of a certain family of local Cohen-Macaulay rings and obtain, as a corollary, the rationality of the Poincaré series of d-dimensional local Gorenstein rings (R, m) of embedding dimension at least e + d – 3, where e is the multiplicity of R. It follows that local Gorenstein rings of multiplicity at most five have rational Poincaré series.

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