On powers of ideals generated by R-sequences in a Noetherian local ring

1973 ◽  
Vol 74 (3) ◽  
pp. 441-444 ◽  
Author(s):  
A. Caruth
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Primary Decomposition ◽  
Finite Intersection ◽  
Noetherian Local Ring ◽  
Powers Of Ideals ◽  
Primary Ideals ◽  
Unique Maximal Ideal

In a Noetherian commutative ring with identity, every ideal can be expressed (not necessarily uniquely) as a finite intersection of primary ideals (called a primary decomposition). This note is concerned with powers of ideals generated by subsets of an R-sequence in a local ring R (i.e. a Noetherian commutative ring R with identity possessing a unique maximal ideal m) and with a decomposition of such ideals.

scholarly journals Notes on irreducible ideals

1985 ◽  
Vol 31 (3) ◽  
pp. 321-324
Author(s):  
David J. Smith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Primary Ideal ◽  
Regular Local Ring ◽  
Finite Intersection ◽  
Irreducible Components ◽  
Primary Ideals

Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an M-primary ideal of a local ring (R, M) satisfying the condition that Q: M = Q + Ms−1 where s is the index of Q, then all irreducible components of Q have index s. (Q is “index-unmixed”.) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples.

Projective Ideals of Finite Type

1969 ◽  
Vol 21 ◽  
pp. 1057-1061 ◽  
Author(s):  
William W. Smith
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Finite Type ◽  
Local Ring ◽  
Maximal Ideal ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Unique Maximal Ideal

The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used.Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC.Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RPis the quotient ring formed using the complement of P.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Toward a theory of generalized Cohen-Macaulay modules

1986 ◽  
Vol 102 ◽  
pp. 1-49 ◽  
Author(s):  
Ngô Viêt Trung
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Finitely Generated ◽  
Noetherian Local Ring

Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = dim M≥1.

scholarly journals Weak Formal Schemes

1972 ◽  
Vol 45 ◽  
pp. 1-38 ◽  
Author(s):  
David Meredith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Local Ring ◽  
Formal Schemes

Throughout this paper, (R, m) denotes a (noetherian) local ring R with maximal ideal m.In [5], Monsky and Washnitzer define weakly complete R-algebras with respect to m. In brief, an R-algebra A† is weakly complete if

scholarly journals POWERS OF THE MAXIMAL IDEAL AND VANISHING OF (CO)HOMOLOGY

2020 ◽  
Vol 63 (1) ◽  
pp. 1-5
Author(s):  
OLGUR CELIKBAS ◽  
RYO TAKAHASHI
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Positive Power ◽  
Noetherian Local Ring ◽  
Torsion Condition

AbstractWe prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

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].

Artinian modules over commutative rings

1992 ◽  
Vol 111 (1) ◽  
pp. 25-33 ◽  
Author(s):  
R. Y. Sharp
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Matlis Duality ◽  
Artinian Module

In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

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 On the Cohen-Macaulayfication of certain Buchsbaum rings

1980 ◽  
Vol 80 ◽  
pp. 107-116 ◽  
Author(s):  
Shiro Goto
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Local Ring ◽  
Parameter Ideal ◽  
The Difference ◽  
System Of Parameters

Let A be a Noetherian local ring of dimension d and with maximal ideal m. Then A is called Buchsbaum if every system of parameters is a weak sequence. This is equivalent to the condition that, for every parameter ideal q, the difference is an invariant I(A) of A not depending on the choice of q. (See Section 2 for the detail.) The concept of Buchsbaum rings was introduced by Stückrad and Vogel [8], and the theory of Buchsbaum singularities is now developing (cf. [6], [7], [9], [10], and [12]).

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