On equimultiplicity

1982 ◽  
Vol 91 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Herrmann ◽  
U. Orbanz
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Analytic Spread

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

scholarly journals A note on the invertible ideal theorem

1984 ◽  
Vol 25 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Andy J. Gray
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Simple Proof ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Central Element ◽  
Commutative Case ◽  
Additional Information ◽  
The Subject

This note is devoted to giving a conceptually simple proof of the Invertible Ideal Theorem [1, Theorem 4·6], namely that a prime ideal of a right Noetherian ring R minimal over an invertible ideal has rank at most one. In the commutative case this result may be easily deduced from the Principal Ideal Theorem by localizing and observing that an invertible ideal of a local ring is principal. Our proof is partially analogous in that it utilizes the Rees ring (denned below) in order to reduce the theorem to the case of a prime ideal minimal over an ideal generated by a single central element, which can be easily dealt with by adapting the commutative argument in [8]. The reader is also referred to the papers of Jategaonkar on the subject [5, 6, 7], particularly the last where another proof of the theorem appears which yields some additional information.

The Étale locus in complete local rings

2021 ◽  
pp. 49-62
Author(s):  
Raymond Heitmann
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Content Type ◽  
Complete Local Ring

Let R R be a complete local ring and let Q Q be a prime ideal of R R . It is determined precisely which conditions on R R are equivalent to the existence of a complete unramified regular local ring A A and an element g ∈ A − Q g\in A-Q such that R R is a finite A A -module and A g ⟶ R g A_g\longrightarrow R_g is étale . A number of other properties of the possible embeddings A ⟶ R A\longrightarrow R are developed in the process including the determination of precisely which fields can be coefficient fields in the Cohen-Gabber Theorem.

Recursive elements and constructive extensions of computable local integral domains

1973 ◽  
Vol 38 (2) ◽  
pp. 272-290 ◽  
Author(s):  
Glen H. Suter
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Algebraic Structures ◽  
Computable Structures ◽  
Integral Domains ◽  
Local Integral ◽  
Algebraic Operations ◽  
Axiom Systems

With reservations, one can think of abstract algebra as the study of what consequences can be drawn from the axioms associated with certain concrete algebraic structures. Two important examples of such concrete algebraic structures are the integers and the rational numbers. The integers and the rational numbers have two properties which are not in general mirrored in the abstract axiom systems associated with them. That is, the integers and the rational numbers both have effectively computable metrics and their algebraic operations are effectively computable. The study of abstract algebraic systems which possess effectively computable algebraic operations has produced many interesting results. One can think of a computable algebraic structure as one whose elements have been labeled by the set of positive integers and whose operations are effectively computable. The formal definition of computable local integral domain will be given in §3. Some specific computable structures which have been studied are the integers, the rational numbers, and the rational numbers with p-adic valuation. Computable structures were studied in general by Rabin [12]. This paper concerns computable local integral domains as exemplified by the local integral domain Zp. Zp is the localization of the integers with respect to the maximal prime ideal generated by the positive prime p. We should note that the concept of local integral domain is not first order.Let the ordered pair (Q, M) stand for a local ring, where Q is the local ring and M is the unique maximal prime ideal of Q. Since most of my results are proving the existence of certain effective procedures, the assumption that Q has a principal maximal ideal M (rather than M has n generators) greatly simplifies many of the proofs.

scholarly journals The asymptotic nature of the analytic spread

1979 ◽  
Vol 86 (1) ◽  
pp. 35-39 ◽  
Author(s):  
M. Brodmann
Keyword(s):  
Local Ring ◽  
Result Section ◽  
Easy Consequence ◽  
Noetherian Local Ring ◽  
Section 8 ◽  
Asymptotic Nature ◽  
Analytic Spread ◽  
Image Position ◽  
Almost All ◽  
Local Nature

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).

Note on analytic spread and asymptotic sequences

1983 ◽  
Vol 93 (1) ◽  
pp. 49-55 ◽  
Author(s):  
L. J. Ratliff
Keyword(s):  
Local Ring ◽  
Prime Divisor ◽  
New Approaches ◽  
Analytic Spread ◽  
Minimal Prime ◽  
Equality Holding

Since the foundational paper (10) by Northcott and Rees in 1954 there have been quite a few papers concerning reductions of ideals and the analytic spread of an ideal. One particular line of investigation concerning the analytic spread l(I) of an ideal I in a local ring (R, M) was begun in 1972 by Burch in (5), where it was shown that l(I) ≤ altitude R – min (grade R/In; n ≥ 1). This result was sharpened in 1980–81 by Brodmann in three papers, (2, 3, 4). Therein he showed that the sets {grade R/In; n ≥ 1} and {grade In−1/In; n ≥ 1} stabilize for all large n, and calling the stable values t and t*, respectively, it holds that t ≤ t* and l(I) ≤ altitude R – t* when I is not nilpotent. He then gave a case (involving R being quasi-unmixed) when equality holds. In 1981 in (20) Rees used two new approaches to Burch's inequality, and he proved two nice results which may both be stated as: l(I) ≤ altitude R – s(I) with equality holding when R is quasi-unmixed; here, s(I) = min {height P; P is a minimal prime divisor of (M, u) R[tI, u]}– 1 (in the first theorem), and s(I) is the length of a maximal asymptotic sequence over I (in the second theorem).

PROPERTIES OF K-RINGS AND RINGS SATISFYING SIMILAR CONDITIONS

2011 ◽  
Vol 21 (08) ◽  
pp. 1381-1394 ◽  
Author(s):  
CHANG IK LEE ◽  
YANG LEE
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Jacobson Radical ◽  
Quotient Ring ◽  
Basic Structure ◽  
Zero Divisors ◽  
Commutative Domain

Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstein into the commutativity of rings. In this note we focus on the ring-theoretic properties of K-rings. We first construct basic examples of K-rings to be handled easily. It is shown that a semiprime K-ring of bounded index of nilpotency is a commutative domain. It is proved that if R is a prime K-ring then its classical quotient ring is a local ring with a nil Jacobson radical. We also show that if R is a π-regular K-ring then R/P is a field for every strongly prime ideal P of R. The basic structure of a condition, unifying K-rings and reversible rings, is studied with respect to zero-divisors in matrices and polynomials.

scholarly journals Equimultiple coefficient ideals

2017 ◽  
Vol 121 (1) ◽  
pp. 5 ◽  
Author(s):  
P. H. Lima ◽  
V. H. Jorge Pérez
Keyword(s):  
Local Ring ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Analytic Spread ◽  
Minimal Prime ◽  
Primary Ideals

Let $(R,\mathfrak {m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,\dots ,s\}$. The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p} })=e_{i}(L_{\mathfrak{p} })$ for each $i \in \{1,\dots ,k\}$ and each minimal prime $\mathfrak{p} $ of $I$ is called the $k$-th equimultiple coefficient ideal denoted by $I_{k}$. It is a generalization of the coefficient ideals introduced by Shah for the case of $\mathfrak {m}$-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring $G_{I}(R)$ satisfies the $S_{1}$ condition if and only if $I^{n}=(I^{n})_{1}$ for all $n$.

On coefficient ideals

2018 ◽  
Vol 167 (02) ◽  
pp. 285-294 ◽  
Author(s):  
R. CALLEJAS-BEDREGAL ◽  
V. H. JORGE PÉREZ ◽  
M. DUARTE FERRARI
Keyword(s):  
Local Ring ◽  
Structure Theorem ◽  
Noetherian Local Ring ◽  
Analytic Spread ◽  
A Chain ◽  
Primary Ideals

AbstractLet (R, 𝔪) be a Noetherian local ring and I an arbitrary ideal of R with analytic spread s. In [3] the authors proved the existence of a chain of ideals I ⊆ I[s] ⊆ ⋅⋅⋅ ⊆ I[1] such that deg(PI[k]/I) < s − k. In this article we obtain a structure theorem for this ideals which is similar to that of K. Shah in [10] for 𝔪-primary ideals.

A Note on Buchsbaum Rings and Localizations of Graded Domains

1980 ◽  
Vol 32 (5) ◽  
pp. 1244-1249 ◽  
Author(s):  
U. Daepp ◽  
A. Evans
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Integral Domain ◽  
Multiplicative System ◽  
Gorenstein Rings ◽  
Image Position ◽  
The Relationship ◽  
Graded Integral Domain ◽  
Mild Restriction

Let R = ⊕i ≧0Ri be a graded integral domain, and let p ∈ Proj (R) be a homogeneous, relevant prime ideal. Let R(p) = {r/t| r ∈ Ri, t ∈ Ri\p} be the geometric local ring at p and let Rp = {r/t| r ∈ R, t ∈ R\p} be the arithmetic local ring at p. Under the mild restriction that there exists an element r1 ∈ R1\p, W. E. Kuan [2], Theorem 2, showed that r1 is transcendental over R(p) andwhere S is the multiplicative system R\p. It is also demonstrated in [2] that R(p) is normal (regular) if and only if Rp is normal (regular). By looking more closely at the relationship between R(p) and R(p), we extend this result to Cohen-Macaulay (abbreviated C M.) and Gorenstein rings.

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