scholarly journals Completely pseudo-valuation rings and their extensions

2014 ◽  
Vol 95 (109) ◽  
pp. 249-254
Author(s):  
Vijay Bhat
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Valuation Ring ◽  
Rational Numbers ◽  
Commutative Noetherian Ring ◽  
Valuation Rings ◽  
Minimal Prime

Recall that a commutative ring R is said to be a pseudo-valuation ring if every prime ideal of R is strongly prime. We define a completely pseudovaluation ring. Let R be a ring (not necessarily commutative). We say that R is a completely pseudo-valuation ring if every prime ideal of R is completely prime. With this we prove that if R is a commutative Noetherian ring, which is also an algebra over Q (the field of rational numbers) and ? a derivation of R, then R is a completely pseudo-valuation ring implies that R[x, ?] is a completely pseudo-valuation ring. We prove a similar result when prime is replaced by minimal prime.

scholarly journals On the vanishing and the positivity of intersection multiplicities over local rings with small non complete intersection loci

1994 ◽  
Vol 136 ◽  
pp. 133-155 ◽  
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Prime Ideal ◽  
Local Rings ◽  
Regular Local Ring ◽  
Commutative Noetherian Ring ◽  
Intersection Multiplicities ◽  
Minimal Prime

Throughout this paperAis a commutative Noetherian ring of dimensiondwith the maximal ideal m and we assume that there exists a regular local ringSsuch thatAis a homomorphic image ofS, i.e.,A=S/Ifor some idealIofS. Furthermore we assume thatAis equi-dimensional, i.e., dimA= dimA/for any minimal prime idealofA. We put.

Positive deissler rank and the complexity of injective modules

1988 ◽  
Vol 53 (1) ◽  
pp. 284-293 ◽  
Author(s):  
T. G. Kucera
Keyword(s):  
Prime Ideal ◽  
Lower Bounds ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Primary Ideals

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

scholarly journals Artinian and Non-Artinian Local Cohomology Modules

2011 ◽  
Vol 54 (4) ◽  
pp. 619-629 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Alireza Vahidi
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Maximal Element ◽  
Local Cohomology ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Finite Module ◽  
Bass Numbers ◽  
Cohomology Modules

AbstractLet M be a finite module over a commutative noetherian ring R. For ideals a and b of R, the relations between cohomological dimensions of M with respect to a, b, a ⋂ b and a + b are studied. When R is local, it is shown that M is generalized Cohen–Macaulay if there exists an ideal a such that all local cohomology modules of M with respect to a have finite lengths. Also, when r is an integer such that 0 ≤ r < dimR(M), any maximal element q of the non-empty set of ideals ﹛a : (M) is not artinian for some i, i ≥ r} is a prime ideal, and all Bass numbers of (M) are finite for all i ≥ r.

scholarly journals On asymptotic values of certain sets of attached prime ideals

1988 ◽  
Vol 30 (3) ◽  
pp. 293-300 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Prime Divisors ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Asymptotic Values ◽  
Attached Prime ◽  
Positive Integers

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

scholarly journals On the Diameter of Dual Graphs of Stanley-Reisner Rings and Hirsch Type Bounds on Abstractions of Polytopes

10.37236/6831 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Brent Holmes
Keyword(s):  
Noetherian Ring ◽  
Dual Graph ◽  
Upper Bounds ◽  
Maximum Diameter ◽  
Prime Ideals ◽  
Lower And Upper Bounds ◽  
Commutative Noetherian Ring ◽  
Positive Dimension ◽  
Dual Graphs ◽  
Minimal Prime

Let $R$ be an equidimensional commutative Noetherian ring of positive dimension. The dual graph $\mathcal{G} (R)$ of $R$ is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height$(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of dual graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of variables and the dimension. Dual graphs of $(S_2)$ Stanley-Reisner rings are a natural abstraction of the $1$-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type bounds on $1$-skeletons of polyhedra.

scholarly journals Modules with minimax Cousin cohomologies

2020 ◽  
Vol 30 (1) ◽  
pp. 143-149
Author(s):  
A. Vahidi ◽  
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cohomology Modules ◽  
Cousin Complex

Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X, the height of p: (i) the nth Bass number of X with respect to p is finite; (ii) the nth local cohomology module of Xp with respect to pRp is Artinian.

On Maximal Torsion Radicals

1973 ◽  
Vol 25 (4) ◽  
pp. 712-726 ◽  
Author(s):  
John A. Beachy
Keyword(s):  
General Result ◽  
Noetherian Ring ◽  
Associative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Maximal Torsion ◽  
Minimal Prime

Let R be an associative ring with identity, and let denote the category of unital left R-modules. It is known that if R is a commutative, Noetherian ring, then the maximal torsion radicals of correspond to the minimal prime ideals of R. In fact, Nӑstӑsescu and Popescu [15, Proposition 2.7] have given a more general result valid for arbitrary commutative rings. This paper investigates maximal torsion radicals over rings not necessarily commutative.

The Artin–Rees equations

1987 ◽  
Vol 102 (3) ◽  
pp. 385-387
Author(s):  
A. Caruth
Keyword(s):  
Prime Ideal ◽  
Unique Solution ◽  
Noetherian Ring ◽  
Identity Element ◽  
Solution Set ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Maximal Force ◽  
The Family ◽  
Image Position

Let R denote a commutative Noetherian ring with an identity element and N a finitely generated R -module. When K is a submodule of N and A an ideal of R the Artin–Rees lemma states that there is an integer q ≥ 0 such that AnN ∩ K = An−q(AqN ∩ K) for all n ≥ q (Rees[4]; Northcott [3], theorem 20, p. 210; Atiyah and Macdonald [1], proposition 10·9, p. 107; Nagata [2], theorem (3·7), p. 9). The above equation belongs to the family of module equations involving A and K which is considered below. We characterize, in terms of A and K, the set of submodules X of N for which there is an integer q = q(X) ≥ 0 satisfying the equationEquation (1), which we call the Artin–Rees equation related to A and K, gets its maximal force when X is largest and we determine the best possible solution in this sense. Notice that for any submodule X satisfying (1), X ⊆ K:NAn for all n ≥ q(X). Since N is a Noetherian R-module ([3], proposition 1 (corollary), p. 177), there is an integer t ≥ 1 such that K:NAt = K:NAt+n for all n ≥ 0. We define M = K:NAt and prove, in Theorem 1, that X = Q satisfies equation (1), for a suitable integer q(Q) ≥ 0, if and only if K ⊆ Q:NAυ ⊆ M for some integer υ ≥ 0. In topological terms, the A-adic topology of K coincides with the topology induced by the A-adic topology of N on the subspace Q if the inequality K ⊆ Q:NAυ ⊆ M is satisfied. It follows that the solution set of equation (1) includes every submodule of N of the form An−rK when n ≥ r = q(K) as well as every submodule lying between K and M. Hence, X = M is the strongest solution, in the sense that M is the largest such submodule contained in An−s (AsN ∩ K): NAn for all n ≥ s = q(M). Recall that M is strictly larger than K if and only if A is contained in at least one prime ideal of R belonging to K ([3], theorem 14 (corollary 1), p. 193). Thus, equation (1) has a unique solution (necessarily X = K) if and only if A is not contained in any prime ideal of R belonging to any solution.

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