Prime Ideals Research Articles

Banaschewski compactifications via special rings of functions in the absence of the Axiom of Choice

For a topological space is the ring of all continuous real functions f on X such that, for every real number ϵ > 0, there exists a countable clopen cover of X such that the oscillation of f on each member of is less than ϵ. For a zero-dimensional T 1-space X, the ring and its subring of bounded functions from are applied to necessary and sufficient conditions on X to admit the Banaschewski compactification in the absence of the Axiom of Choice. For a zero-dimensional T 1-space X and a Tychonoff space Y, the problem of when the ring can be isomorphic to or to the ring of all (bounded) continuous real functions on Y is investigated. Several new equivalences of the Boolean Prime Ideal Theorem are deduced. Some results about are obtained under the Principle of Countable Multiple Choices.

Some Results Concerning Uniformly (1-Absorbing) Primary Ideals

Some Results Concerning Uniformly (1-Absorbing) Primary Ideals

On Energy of Prime Ideal Graph of A Commutative Ring Associated with Seidel-Based Matrices

Some energies of the prime ideal graph are found for a commutative ring associated with Seidel-based matrices including Seidel, Seidel Laplacian, and Seidel signless Laplacian matrices.

Some remarks on S -Noetherian modules

UDC 512.5 We study several properties and applications of the S -Noetherian rings and modules. It is proved that an S -Artinian ring is S -Noetherian provided that S contains no zero divisors of the module. Furthermore, it is shown that associated primes exist in modules over the S -Noetherian rings and the major part of notions of associated prime ideals coincide over the S -Noetherian rings. We also extend the classical Krull's intersection theorem for S -Noetherian rings. Moreover, we provide a characterization of the S -Noetherian modules in terms of the G -graded S -Noetherian modules.

Constructing noncatenary quasi-excellent precompletions

Let T be a local (Noetherian) ring and let Q 1 and Q 2 be prime ideals of T. We find sufficient conditions for there to exist a quasi-excellent local subring B of T satisfying the following conditions: (1) the completion of B at its maximal ideal is isomorphic to the completion of T at its maximal ideal, (2) B ∩ Q 1 = B ∩ Q 2 , (3) the set of prime ideals of T / ( Q 1 ∩ Q 2 ) of positive height is the same as the set of prime ideals of B / ( B ∩ Q 1 ) of positive height when viewed as partially ordered sets, and (4) for i = 1 and for i = 2, there is a coheight preserving bijection between the minimal prime ideals of T Q i and the minimal prime ideals of B B ∩ Q 1 . Intuitively, this means that T contains a quasi-excellent local subring in which Q 1 and Q 2 are “glued together” and such that both the completion and desirable properties of the prime spectrum are preserved. We use this result to show that certain complete local rings are the completion of a quasi-excellent local ring whose prime spectrum, when viewed as a partially ordered set, contains interesting noncatenary finite subsets.

A topology based on collections of prime ideals disjoint from a multiplicatively closed set

In this article, we study a topology on the prime spectrum of a ring that is finer than the Zariski topology. For a commutative ring R with identity, we introduce Spec ˜ ( R ) as a topological space whose underlying set is the prime spectrum Spec ( R ) of R, with basic open sets of the form D ˜ ( S ) : = { p ∈ Spec ( R ) | p ∩ S = ∅ } , where S is a multiplicatively closed subset of R. Various topological properties of Spec ˜ ( R ) are investigated. For example, we show that Spec ˜ ( R ) is locally compact, locally contractible, and coherent with the family of all its finite subspaces. Moreover, we provide equivalent conditions for the connectedness, separability, Lindelöfness, regularity, and submaximality of Spec ˜ ( R ) .

A Study of Bipolar Fuzzy Prime Ideals of a Lattice

A Study of Bipolar Fuzzy Prime Ideals of a Lattice

Controlling formal fibers of countably many principal prime ideals

Let [Formula: see text] be a complete local (Noetherian) ring. For each [Formula: see text], let [Formula: see text] be a nonempty countable set of nonmaximal, pairwise incomparable prime ideals of [Formula: see text], and suppose that if [Formula: see text], then either [Formula: see text] or no element of [Formula: see text] is contained in an element of [Formula: see text]. We provide necessary and sufficient conditions for [Formula: see text] to be the completion of a local integral domain [Formula: see text] satisfying the condition that, for all [Formula: see text], there is a nonzero prime element [Formula: see text] of [Formula: see text] such that [Formula: see text] is exactly the set of maximal elements of the formal fiber of [Formula: see text] at [Formula: see text]. We then prove related results where the domain [Formula: see text] is required to be countable and/or excellent.

A Study of Generalized Differential Identities via Prime Ideals

Let be a ring and be a prime ideal of . The aim of this research paper is to delve into the relationship between the structural properties of the quotient ring and the behavior of generalized derivations in a ring endowed with an involution. Precisely, the study focuses on characterizing generalized derivations in the ring that are associated with prime ideals and satisfy certain functional identities. The investigation seeks to uncover how these derivations interact with the quotient ring structure and the implications for the underlying ring theory. The key objective is to explore the relationship between a structure and a map, a topic that has proven to be highly significant. Moreover, we offer an example to show that the numerous conditions outlined in the hypotheses of our theorems are reasonable and not overly stringent.

Ideals and filters on intuitionistic fuzzy lattices

Based on the concept of Atanassov’s intuitionistic fuzzy set on a universe X, we introduce the concepts of intuitionistic fuzzy ideals and intuitionistic fuzzy filters on an intuitionistic fuzzy lattice. More specifically, we provide characterizations of these concepts in terms of the intuitionistic fuzzy lattice meet and join operations, in terms of some associated fuzzy sets, as well as, in terms of their crisp level sets. Furthermore, we introduce the concepts of prime intuitionistic fuzzy ideals (resp. filters) as interesting kinds, and investigate their various properties and characterizations.

A study of 1-absorbing intuitionistic fuzzy ideals of commutative semirings

The primary objective of this investigation is to offer an extensive and thorough comprehension of the characteristics, attributes, and operational tendencies inherent in 1-absorbing intuitionistic fuzzy ideals as well as 1-absorbing intuitionistic fuzzy primary ideals within the context of commutative semirings. By examining their contribution to the literature, we can gain insight into their unique characteristics and explore their potential applications in various fields. Furthermore, defining the Cartesian product of these ideals allows for a deeper analysis of their interactions and opens up new avenues for research in this area.

A Generalization of Source of Semiprimeness

This paper characterizes the semigroup ideal $\mathcal{L}_{R}^{n}(I)$ of a ring $R$, where $I$ is an ideal of $R$, defined by $\mathcal{L}_{R}^{0}(I)=I$ and $\mathcal{L}_{R}^{n}(I)=\{a\in R \mid aRa\subseteq \mathcal{L}_{R}^{n-1}(I)\}$, for all $n\in \mathbb{Z}^+$, the set of all the positive integers. Moreover, it studies the basic properties of the set $\mathcal{L}_{R}^{n}(I)$ and defines $n$-prime ideals, $n$-semiprime ideals, $n$-prime rings, and $n$-semiprime rings. This study also investigates relationships between the sets $\mathcal{L}_{R}(I)$ and $\mathcal{L}_{R}^{n}(I)$ and exemplifies some of the related properties. It obtains the main results concerning prime rings and prime ideals by the properties of the set $\mathcal{L}_{R}^{n}(I)$.

A note on (m, n)-absorbing primary ideals of commutative rings

A note on (m, n)-absorbing primary ideals of commutative rings

Strongly S-1-Absorbing Primary Ideals of Commutative Rings

Strongly S-1-Absorbing Primary Ideals of Commutative Rings

Equivariant algebraic and semi-algebraic geometry of infinite affine space

We study Sym(∞)-orbit closures of non-necessarily closed points in the Zariski spectrum of the infinite polynomial ring C[xij:i∈N,j∈[n]]. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by Sym(∞) orbits of polynomials in R[xij:i∈N,j∈[n]]. For n=1 we prove a quantifier elimination type result which fails for n>1.

On Sums Over Prime Ideals

On Sums Over Prime Ideals

Isomorphism Problems and Groups of Automorphisms for Ore Extensions K[x][y;fddx] (Prime Characteristic)

Let Λ(f)=K[x][y;fddx] be an Ore extension of a polynomial algebra K[x] over an arbitrary field K of characteristic p>00$$\\end{document}]]> where f∈K[x]. For each polynomial f, the automorphism group of the algebras Λ(f) is explicitly described. The automorphism group AutK(Λ(f))=S⋊Gf is a semidirect product of two explicit groups where Gf is the eigengroup of the polynomial f (the set of all automorphisms of K[x] such that f is their common eigenvector). For each polynomial f, the eigengroup Gf is explicitly described. It is proven that every subgroup of AutK(K[x]) is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra Λ(f) are 2. The prime, completely prime, primitive and maximal ideals of the algebra Λ(f) are classified.

On weakly S-ρ-ideals in noncommutative rings

Let R be a noncommutative ring, and φ ≠ S ⊆ R be an m-system of R. In this paper, we extend the concept of weakly prime ideals in noncommutative rings by introducing weakly S-ρ-ideals. This generalization builds on the notions of right S-prime, weakly S-n-ideals and S-ρ-ideals. We present equivalent definitions and key properties of weakly S-ρ-ideals. We also explore with the method of idealization how the weakly S-ρ-ideal property transfers to related rings and construct such ideals. On the other hand, we give a generalization of S-ρ-ideals which were introduced in [12].

Stabilization of associated prime ideals of monomial ideals – Bounding the copersistence index

The sequence (Ass(R/In))n∈N of associated primes of powers of a monomial ideal I in a polynomial ring R eventually stabilizes by a known result by Markus Brodmann. Lê Tuân Hoa gives an upper bound for the index where the stabilization occurs. This bound depends on the generators of the ideal and is obtained by separately bounding the powers of I after which said sequence is non-decreasing and non-increasing, respectively. In this paper, we focus on the latter and call the smallest such number the copersistence index. We take up the proof idea of Lê Tuân Hoa, who exploits a certain system of inequalities whose solution sets store information about the associated primes of powers of I. However, these proofs are entangled with a specific choice for the system of inequalities. In contrast to that, we present a generic ansatz to obtain an upper bound for the copersistence index that is uncoupled from this choice of the system. We establish properties for a system of inequalities to be eligible for this approach to work. We construct two suitable inequality systems to demonstrate how this ansatz yields upper bounds for the copersistence index and compare them with Hoa's. One of the two systems leads to an improvement of the bound by an exponential factor.

A characterization of ramification groups via jet algebras

We present a functorial method to define ramification groups, identifying them as inertia groups of an induced action on composite jet algebras. This framework lays the foundation for defining higher ramification groups for actions involving group schemes. To achieve this, we introduce Taylor maps within the category of commutative unitary rings at prime ideals of an R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R$$\\end{document}-algebra and compute their kernels for algebras of finite type over a field with separably generated residue fields.