Noetherian orders

2010 ◽  
Vol 21 (1) ◽  
pp. 111-124 ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Noetherian Ring ◽  
Constructive Proof ◽  
Finitely Generated ◽  
Hilbert Basis ◽  
Basis Theorem ◽  
Natural Way

Noether classes of posets arise in a natural way from the constructively meaningful variants of the notion of a Noetherian ring. Using an axiomatic characterisation of a Noether class, we prove that if a poset belongs to a Noether class, then so does the poset of the finite descending chains. When applied to the poset of finitely generated ideals of a ring, this helps towards a unified constructive proof of the Hilbert basis theorem for all Noether classes.

scholarly journals A Difference-Differential Basis Theorem

1970 ◽  
Vol 22 (6) ◽  
pp. 1224-1237 ◽  
Author(s):  
Richard M. Cohn
Keyword(s):  
Commutative Ring ◽  
Chain Condition ◽  
Finitely Generated ◽  
Hilbert Basis ◽  
Radical Ideal ◽  
Differential Basis ◽  
Basis Theorem ◽  
Finite Set ◽  
Radical Ideals ◽  
Ascending Chain Condition

Our aim in this paper is to extend to difference-differential rings the beautiful theorem of Kolchin [5, Theorem 3] for the differential case. The necessity portion of Kolchin's result is not obtained.What might well be called the Ritt basis theorem states that if a commutative ring R with identity is finitely generated over a subring R0, then the ascending chain condition for radical ideals of R0 implies the ascending chain condition for radical ideals of R. (This is indeed a basis theorem. If we define a basis for a radical ideal A to be a finite set B such that then every radical ideal of a ring R has a basis if and only if the ascending chain condition for radical ideals holds in R.) It is the Ritt basis theorem rather than the Hilbert basis theorem which has appropriate generalizations in differential and difference algebra, where in fact it originated.

scholarly journals Constructing Gröbner bases for Noetherian rings

2013 ◽  
Vol 24 (2) ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Gröbner Basis ◽  
Finite Depth ◽  
Commutative Rings ◽  
Polynomial Ideal ◽  
Constructive Proof ◽  
Noetherian Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Prime Decomposition ◽  
Separate Paper

We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

Artinianness of Composed Graded Local Cohomology Modules

2016 ◽  
Vol 59 (2) ◽  
pp. 271-278
Author(s):  
Fatemeh Dehghani-Zadeh
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Primary Ideal ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Base Ring ◽  
Cohomology Modules ◽  
Local Base

AbstractLet be a graded Noetherian ring with local base ring (R0 ,m0) and let . Let M and N be finitely generated graded R-modules and let a = a0 + R+ an ideal of R. We show that and are Artinian for some i s and j s with a specified property, where bo is an ideal of R0 such that a0 + b0 is an m0-primary ideal.

Ideals with Trivial Conormal Bundle

1980 ◽  
Vol 32 (1) ◽  
pp. 210-218 ◽  
Author(s):  
A. V. Geramita ◽  
C. A. Weibel
Keyword(s):  
Polynomial Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Question Answering ◽  
General Theorem ◽  
Closed Field ◽  
Natural Question ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Conormal Bundle

Throughout this paper all rings considered will be commutative, noetherian with identity. If R is such a ring and M is a finitely generated R-module, we shall use v(M) to denote that non-negative integer with the property that M can be generated by v(M) elements but not by fewer.Since every ideal in a noetherian ring is finitely generated, it is a natural question to ask what v(I) is for a given ideal I. Hilbert's Nullstellensatz may be viewed as the first general theorem dealing with this question, answering it when I is a maximal ideal in a polynomial ring over an algebraically closed field.More recently, it has been noticed that the properties of an R-ideal I are intertwined with those of the R-module I/I2.

Some properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (06) ◽  
pp. 1950113 ◽  
Author(s):  
Elham Tavasoli
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Flat Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a nonzero proper ideal of [Formula: see text]. In this paper, we study the properties of a family of rings [Formula: see text], with [Formula: see text], as quotients of the Rees algebra [Formula: see text], when [Formula: see text] is a semidualizing ideal of Noetherian ring [Formula: see text], and in the case that [Formula: see text] is a flat ideal of [Formula: see text]. In particular, for a Noetherian ring [Formula: see text], it is shown that if [Formula: see text] is a finitely generated [Formula: see text]-module, then [Formula: see text] is totally [Formula: see text]-reflexive as an [Formula: see text]-module if and only if [Formula: see text] is totally reflexive as an [Formula: see text]-module, provided that [Formula: see text] is a semidualizing ideal and [Formula: see text] is reducible in [Formula: see text]. In addition, it is proved that if [Formula: see text] is a nonzero flat ideal of [Formula: see text] and [Formula: see text] is reducible in [Formula: see text], then [Formula: see text], for any [Formula: see text]-module [Formula: see text].

scholarly journals On the length of the powers of systems of parameters in local ring

1990 ◽  
Vol 120 ◽  
pp. 77-88 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Finitely Generated ◽  
Systems Of Parameters ◽  
System Of Parameters ◽  
The Ideal

Throughout this note, A denotes a commutative local Noetherian ring with maximal ideal m and M a finitely generated A-module with dim (M) = d. Let x1, …, xd be a system of parameters (s.o.p. for short) for M and I the ideal of A generated by x1, …, xd.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

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