scholarly journals On non-commutative regular local rings

1976 ◽  
Vol 17 (2) ◽  
pp. 98-102 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Artinian Ring ◽  
Nilpotent Groups ◽  
Global Dimension ◽  
Prime Ideals ◽  
Homological Dimension ◽  
Local Rings ◽  
Universal Enveloping Algebras ◽  
Regular Local Rings

Let R be a ring (with identity). We shall call R a local ring if R is aright noetherian ring such that the Jacobson radical M is a maximal ideal (and so is the only maximal ideal), and R/M is a simple artinian ring. A local ring R with maximal ideal M is called regular if there exists a chainof ideals Mi of such that Mi–1/Mi is generated by a central regular element of R/Mi (1 ≦ i ≦ n). For such a ring R, Walker [6, Theorem 2. 7] proved that R is prime and n is the right global dimension of R, the Krull dimension of R, the homological dimension of theR-module R/M and the supremum of the lengths of chains of prime ideals of R. Such regular local rings will be called n-dimensional. The aim of this note is to give examples of regular local rings. These arise as localizations of universal enveloping algebras of nilpotent Lie algebras over fields and localizations of group algebras of certain finitely generated finite-by-nilpotent groups.

On ideals of finite homoloǵical dimension in local rings

1968 ◽  
Vol 64 (4) ◽  
pp. 941-948 ◽  
Author(s):  
Lindsay Burch
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Additional Condition ◽  
Homological Dimension ◽  
Local Rings ◽  
Algebraic Properties ◽  
Dimension One

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)

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].

scholarly journals Global dimensions of right coherent rings with left Krull dimension

1989 ◽  
Vol 39 (2) ◽  
pp. 215-223 ◽  
Author(s):  
Mark L. Teply
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Homological Dimension ◽  
Isomorphism Classes ◽  
Coherent Ring ◽  
Gabriel Dimension ◽  
Coherent Rings

The weak global dimension of a right coherent ring with left Krull dimension α ≥ 1 is found to be the supremum of the weak dimensions of the β-critical cyclic modules, where β < α. If, in addition, the mapping I → assl gives a bijection between isomorphism classes on injective left R-modules and prime ideals of R, then the weak global dimension of R is the supremum of the weak dimensions of the simple left R-modules. These results are used to compute the left homological dimension of a right coherent, left noetherian ring. Some analogues of our results are also given for rings with Gabriel dimension.

scholarly journals On the Structure of Complete Local Rings

1950 ◽  
Vol 1 ◽  
pp. 63-70 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Structure Theorem ◽  
Finite Basis ◽  
Local Rings ◽  
Topological Ring

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

scholarly journals Joint reductions of complete ideals

1990 ◽  
Vol 118 ◽  
pp. 155-163 ◽  
Author(s):  
J. K. Verma
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Regular Local Rings

The aim of this paper is to extend and unify several results concerning complete ideals in 2-dimensional regular local rings by using the theory of joint reductions and mixed multiplicities. The theory of complete ideals in a 2-dimensional regular local ring was developed by Zariski in his 1938 paper [Z]. This theory is presented in a simpler and general form in [ZS, Appendix 5] and [H2].

scholarly journals On the Unit Graph of a Non-commutative Ring

2015 ◽  
Vol 22 (spec01) ◽  
pp. 817-822 ◽  
Author(s):  
S. Akbari ◽  
E. Estaji ◽  
M.R. Khorsandi
Keyword(s):  
Finite Field ◽  
Commutative Ring ◽  
Bipartite Graph ◽  
Local Ring ◽  
Maximal Ideal ◽  
Complete Bipartite Graph ◽  
Artinian Ring ◽  
Clique Number ◽  
Unit Element ◽  
Unit Graph

Let R be a ring with non-zero identity. The unit graph G(R) of R is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and 𝔪 is a maximal ideal of R such that |R/𝔪| = 2, then G(R) is a complete bipartite graph if and only if (R, 𝔪) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessarily commutative), then G(R) is a complete r-partite graph if and only if (R, 𝔪) is a local ring and r = |R/𝔪| = 2n for some n ∈ ℕ or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U(R) and the clique number of G(R) is finite, then R is a finite ring.

scholarly journals Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

2010 ◽  
Vol 199 ◽  
pp. 95-105 ◽  
Author(s):  
Shiro Goto ◽  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Local Rings ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients

AbstractLet (A,m) be a Noetherian local ring withd= dimA≥ 2. Then, ifAis a Buchsbaum ring, the first Hilbert coefficientsofAfor parameter idealsQare constant and equal towherehi(A)denotes the length of theith local cohomology moduleofAwith respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves thatAis a Buchsbaum ring ifAis unmixed and the valuesare constant, which are independent of the choice of parameter idealsQinA. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

scholarly journals On the Sum of Homological Dimension and Codimension of Modules over a Semi-local Ring

1968 ◽  
Vol 33 ◽  
pp. 165-172
Author(s):  
Samuel S.H. Young
Keyword(s):  
Local Ring ◽  
Global Dimension ◽  
Homological Dimension ◽  
Regular Local Ring ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Arbitrary Ring ◽  
Local Case

Let M be a finitely generated module over a regular local ring R. It is well known that the sum of homological dimension and codimension of M is equal to the global dimension of R. For modules over an arbitrary ring, this is in general not true. The purpose of this paper is to investigate the properties of such sums in the semi-local case.

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).

scholarly journals Noetherian Rings—Dimension and Chain Conditions

2005 ◽  
Vol 4 (3) ◽  
Author(s):  
Abhishek Banerjee
Keyword(s):  
Sufficient Conditions ◽  
Artinian Ring ◽  
Small Dimension ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Local Rings ◽  
Direct Products ◽  
Finite Set ◽  
Minimal Prime ◽  
Necessary And Sufficient

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime ideals of a noetherian ring. In particular, we express noetherian rings with certain properties as finite direct products of noetherian rings with a unique minimal prime ideal, as an analogue to the expression of an artinian ring as a finite direct product of artinian local rings. Besides, we also consider the set of ideals I in R such that M ≠ I M for a given module M and show that a maximal element among these is prime. In Section Four, we deal with dimensions of prime ideals, Krull’s Small Dimension Theorem and generalize it (and its converse) to the case of a finite set of prime ideals. Towards the end of the paper, we also consider the sets of linear dependencies that might hold between the generators of an ideal and consider the ideals generated by the coefficients in such linear relations.

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