scholarly journals THE MINIMAL PRIME IDEAL OF A VALUATION RING

2004 ◽  
Vol 23 (1) ◽  
Author(s):  
ROSALI BRUSAMARELLO ◽  
JOÃO GERÔNIMO ◽  
OSVALDO DO ROCÍO
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Minimal Prime Ideal ◽  
Minimal Prime

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 Primary Decomposition of Ideals of Lattice Homomorphisms

10.37236/7694 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Leila Sharifan ◽  
Ali Akbar Estaji ◽  
Ghazaleh Malekbala
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Minimal Prime Ideal ◽  
Primary Decomposition ◽  
Planar Lattice ◽  
Combinatorial Properties ◽  
Finite Lattices ◽  
Minimal Prime ◽  
Lattice Homomorphisms ◽  
The Ideal

For two given finite lattices $L$ and $M$, we introduce the ideal of lattice homomorphism $J(L,M)$, whose minimal monomial generators correspond to lattice homomorphisms $\phi : L\to M$. We show that $L$ is a distributive lattice if and only if the equidimensinal part of $J(L,M)$ is the same as the equidimensional part of the ideal of poset homomorphisms $I(L,M)$. Next, we study the minimal primary decomposition of $J(L,M)$ when $L$ is a distributive lattice and $M=[2]$. We present some methods to check if a monomial prime ideal belongs to $\mathrm{ass}(J(L,[2]))$, and we give an upper bound in terms of combinatorial properties of $L$ for the height of the minimal primes. We also show that if each minimal prime ideal of $J(L,[2])$ has height at most three, then $L$ is a planar lattice and $\mathrm{width}(L)\leq 2$. Finally, we compute the minimal primary decomposition when $L=[m]\times [n]$ and $M=[2]$.

scholarly journals U–essential prime divisors and sequences over an ideal

1986 ◽  
Vol 103 ◽  
pp. 39-66 ◽  
Author(s):  
Daniel Katz ◽  
Louis J. Ratliff
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Asymptotic Theory ◽  
Minimal Prime Ideal ◽  
Standard Theory ◽  
Noetherian Rings ◽  
Associated Primes ◽  
Prime Divisors ◽  
Minimal Prime

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian.In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, R-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasi-unmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than R-sequences (for example, they behave nicely when passing to R/z with z a minimal prime ideal in R), the converse of this working guide has also proved useful.

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.

On a question of D. Rees on classical integral closure and integral closure relative to an Artinian module

2019 ◽  
Vol 19 (02) ◽  
pp. 2050033
Author(s):  
V. H. Jorge Pérez ◽  
L. C. Merighe
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Local Cohomology ◽  
Integral Closure ◽  
Minimal Prime Ideal ◽  
Local Cohomology Module ◽  
Artinian Module ◽  
Classical Integral ◽  
Minimal Prime ◽  
The Relationship

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] and [Formula: see text] ideals of [Formula: see text]. Motivated by a question of Rees, we study the relationship between [Formula: see text], the classical Northcott–Rees integral closure of [Formula: see text], and [Formula: see text], the integral closure of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text] (also called here ST-closure of [Formula: see text] on [Formula: see text]), in order to study a relation between [Formula: see text], the multiplicity of [Formula: see text], and [Formula: see text], the multiplicity of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text]. We conclude [Formula: see text] when every minimal prime ideal of [Formula: see text] belongs to the set of attached primes of [Formula: see text]. As an application, we show what happens when [Formula: see text] is a generalized local cohomology module.

Annulets and α-ideals in a distributive lattice

1973 ◽  
Vol 15 (1) ◽  
pp. 70-77 ◽  
Author(s):  
William H. Cornish
Keyword(s):  
Boolean Algebra ◽  
Prime Ideal ◽  
Distributive Lattice ◽  
Minimal Prime Ideal ◽  
Distributive Lattices ◽  
Basic Theorem ◽  
Minimal Prime

In a distributive lattice L with 0 the set of all ideals of the form (x]* can be made into a lattice A0(L) called the lattice of annulets of L. A 0(L) is a sublattice of the Boolean algebra of all annihilator ideals in L. While the lattice of annulets is no more than the dual of the so-called lattice of filets (carriers) as studied in the theory of l-groups and abstractly for distributive lattices in [1, section4] it is a useful notion in its own right. For example, from the basic theorem of [3] it follows that A 0(L) is a sublattice of the lattice of all ideals of L if and only if each prime ideal in L contains a unique minimal prime ideal.

Normal lattices

1972 ◽  
Vol 14 (2) ◽  
pp. 200-215 ◽  
Author(s):  
William H. Cornish
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Minimal Prime Ideal ◽  
Minimal Prime

If L is a distributive lattice with 0 then it is shown that each prime ideal contains a unique minimal prime ideal if and only if, for any x and y in L, x ∧ y = 0 implies (x]*) ∨ (y]* L). A distributive lattice with 0 is called normal if it satisfies the conditions of this result. This terminology is appropriate for the following reasons. Firstly the lattice of closed subsets of a T1-space is normal if and only if the space is normal. Secondly lattices satisfying the above annihilator condition are sometimes called normal by those mathematicians interested in (Wallman-) compactications, for example see [2].

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