Prime Ideals Research Articles

Generalization of 2-absorbing quasi primary ideals

Generalization of 2-absorbing quasi primary ideals

On graded 1-absorbing prime ideals

Let G be a group with identity e and R be a G-graded commutative ring with unity 1. In this article, we introduce and study the concept of graded 1-absorbing prime ideals which is a generalization of graded prime ideals. A proper graded ideal P of R is called graded 1-absorbing prime if for all nonunit elements $$x,y,z\in h(R)$$ such that $$xyz\in P$$ , then either $$xy\in P$$ or $$z\in P$$ .

Enlarging localized polynomial rings while preserving their prime ideal structure

Let n be an integer greater than 1 and let x1,…,xn be indeterminates over a countable field k. In this paper, we employ techniques of Heitmann and Nagata to show there exists an uncountable regular local ring S between the localized polynomial ring k[x1,…,xn](x1,…,xn) and the power series ring k[[x1,…,xn]] such that the prime ideal spectrum of S is homeomorphic to the prime ideal spectrum of k[x1,…,xn](x1,…,xn) as topological spaces with the Zariski topology (Theorem 3.17). Thus S is a local n-dimensional Noetherian domain and the cardinality of the set of prime ideals of S is strictly less than the cardinality of S. We also show that every Noetherian ring A with infinitely many prime ideals has a Noetherian subring B such that the prime ideal spectrum of B is homeomorphic to the prime ideal spectrum of A and the cardinality of the set of prime ideals of B equals the cardinality of B.

Variations of primeness and factorization of ideals in Leavitt path algebras

In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under which a proper ideal of a Leavitt path algebra L is a product as well as an intersection of finitely many of these different types of prime ideals. Such factorizations, when they are irredundant, are shown to be unique except for the order of the factors. We also characterize the Leavitt path algebras L in which every ideal admits such factorizations and also in which every ideal is one of these special type of ideals.

Prime Fuzzy Bi-Ideals in Near-Subtraction Semigroups

A study on fuzzy prime ideals in near-subtraction semigroups is already known. We have to expand the concept of prime fuzzy bi-ideals in near-subtraction semigroups and analyse some of its properties to characterize it. This will lead to learn a new type of fuzzy ideal and to develope the researcher to made their research.

Prime Fuzzy Bi-Ideals in Near-Subtraction Semigroups

A study on fuzzy prime ideals in near-subtraction semigroups is already known. We have to expand the concept of prime fuzzy bi-ideals in near-subtraction semigroups and analyse some of its properties to characterize it. This will lead to learn a new type of fuzzy ideal and to develope the researcher to made their research.

COMPLETELY PRIME ONE-SIDED IDEALS IN SKEW POLYNOMIAL RINGS

AbstractLet R = K[x, σ] be the skew polynomial ring over a field K, where σ is an automorphism of K of finite order. We show that prime elements in R correspond to completely prime one-sided ideals – a notion introduced by Reyes in 2010. This extends the natural correspondence between prime elements and prime ideals in commutative polynomial rings.

Primary decomposition in incomplete Noetherian algebras

We show that in a noetherian commutative unital topological algebra, the prime ideals associated with a closed ideal as well as its isolated primary components are closed. We obtain a version of the closed graph theorem. An example of a noetherian (even principal) commutative unital semi-simple and incomplete normed algebra whose each ideal is closed is also given.

A central quaternionic Nullstellensatz

Let I be a proper left ideal in the ring H[x1,…,xn] of polynomials in n central variables over the quaternion algebra H. Then there exists a point a=(a1,…,an)∈Hn with aiaj=ajai for all i,j, such that every polynomial in I vanishes at a. This generalizes a theorem of Jacobson, who proved the case n=1. Moreover, a polynomial f∈H[x1,…,xn] vanishes at all common zeroes of polynomials in I if and only if f belongs to the intersection of all completely prime left ideals that contain I – a notion introduced by Reyes in 2010.

Commutative rings with one-absorbing factorization

Let R be a commutative ring with nonzero identity. Yassine et al. defined the concept of 1-absorbing prime ideals as follows: a proper ideal I of R is said to be a 1-absorbing prime ideal if whenever for some nonunit elements then either or We use the concept of 1-absorbing prime ideals to study those commutative rings in which every proper ideal is a product of 1-absorbing prime ideals (we call them OAF-rings). Any OAF-ring has dimension at most one and local OAF-domains (D, M) are atomic such that M 2 is universal.

Generalised Chain Conditions, Prime Ideals, and Classes of Locally Finite Lie Algebras

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian. Both properties are equivalent to soluble-by-finite. We also prove a structure theorem for serially finite Artinian Lie algebras.

Minimal generators of the ideal class group

We prove unconditionally that in some family of number fields, except for a density zero set, the ideal class group of K is generated by degree one prime ideals whose norm is O((log⁡dK)α) for some α>4, where dK is the absolute value of the discriminant of K, where the implied constant depends only on α.

Primary Ideals and Their Differential Equations

An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE described by a primary ideal amounts to computing Noetherian operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms for this task, and we present efficient implementations.

The Regularity Conjecture for prime ideals in polynomial rings

This paper presents a survey on recent developments on regularity of prime ideals in polynomial rings.

A-minimal prime ideals in almost distributive lattices

a-minimal prime ideals in almost distributive lattices

Characterisation of p-ideals in Hemirings on the Basis of Hemiregularity, Prime and Normal Fuzzy Ideals

In p-hemiregular hemirings, some properties of ‘prime fuzzy p-ideals’ are examined. In the case of fuzzy p-ideal, p-hemiregular hemiring is also characterised. It is showed that in the case of hemiring , fuzzy subset φ is a ‘prime fuzzy left p-ideal’ in if and only if φ is having two values, set having all values of i?> in so that φ(p) equals to one is a ‘prime left(right) p-ideal’ in , φ(0) = 1. In hemirings for ‘maximal fuzzy left(right) p-ideals’, some related properties are also studied.

Spectral maps associated to semialgebraic branched coverings

In this article we prove that a semialgebraic map \(\pi :M\rightarrow N\) is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map has a neat behavior with respect to the branching locus, the ramification set and the ramification index. A crucial fact to prove the preceding result is the characterization of the prime ideals whose fibers under the previous spectral map are singletons.

Ideals on neutrosophic extended triplet groups

<abstract><p>In this paper, we introduce the concept of (prime) ideals on neutrosophic extended triplet groups (NETGs) and investigate some related properties of them. Firstly, we give characterizations of ideals generated by some subsets, which lead to a construction of a NETG by endowing the set consisting of all ideals with a special multiplication. In addition, we show that the set consisting of all ideals is a distributive lattice. Finally, by introducing the topological structure on the set of all prime ideals on NETGs, we obtain the necessary and sufficient conditions for the prime ideal space to become a $ T_{1} $-space and a Hausdorff space. </p></abstract>

On Rough Fuzzy Prime Ideals in Left Almost Semigroups

In this paper we shall introduce the notion of rough prime ideals and rough fuzzy prime ideals in LA-semigroups. We proved that the lower and the upper approximation of a prime ideal is a prime ideal and we also proved that a fuzzy subset f of an LA-semigroup S is a fuzzy prime ideal of S iff fλ≠∅ (fλs≠∅) is a prime ideal of S for every λ∈[0, 1].

Ideal theory on EQ-algebras

<abstract><p>In this article, we introduce ideals and other special ideals on EQ-algebras, such as implicative ideals, primary ideals, prime ideals and maximal ideals. At first, we give the notion of ideal and its related properties on EQ-algebras, and give its equivalent characterizations. We discuss the relations between ideals and filters, and study the generating formula of ideals on EQ-algebras. Moreover, we study the properties of implicative ideals, primary ideals, prime ideals and maximal ideals and their relations. For example, we prove that every maximal ideal is prime and if prime ideals are implicative, then they are maximal in the EQ-algebra with the condition $ (DNP) $. Finally, we introduce the topological properties of prime ideals. We get that the set of all prime ideals is a compact $ T_{0} $ topological space. Also, we transferred the spectrum of EQ-algebras to bounded distributive lattices and given the ideal reticulation of EQ-algebras.</p></abstract>