Prime Ideals Research Articles

On $$\phi $$-1-absorbing prime ideals

In this paper, we introduce $$\phi $$ -1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity $$1\ne 0$$ and $$\phi :\mathcal {I}(R)\rightarrow \mathcal {I}(R)\cup \{\emptyset \}$$ be a function where $$\mathcal {I}(R)$$ is the set of all ideals of R. A proper ideal I of R is called a $$\phi $$ -1-absorbing prime ideal if for each nonunits $$x,y,z\in R$$ with $$xyz\in I-\phi (I)$$ , then either $$xy\in I$$ or $$z\in I$$ . In addition to give many properties and characterizations of $$\phi $$ -1-absorbing prime ideals, we also determine rings in which every proper ideal is $$\phi $$ -1-absorbing prime.

A note on (∈, ∈∨q)-fuzzy prime ideals of near-rings

Using the idea of quasi-coincidence of a fuzzy point with a fuzzy set, we discuss a new kind of (∈, ∈∨q)-fuzzy prime (semiprime) ideals of near-rings. The concept of (∈, ∈∨q 0 δ )-fuzzy prime (semiprime) ideals of near-rings are introduced and some of its related properties are investigated.

Intuitionistic fuzzy normed prime and maximal ideals

<abstract><p>Motivated by the new notion of intuitionistic fuzzy normed ideal, we present and investigate some associated properties of intuitionistic fuzzy normed ideals. We describe the intrinsic product of any two intuitionistic fuzzy normed subsets and show that the intrinsic product of intuitionistic fuzzy normed ideals is a subset of the intersection of these ideals. We specify the notions of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal, we present the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. In addition, the relation between the intuitionistic characteristic function and prime and maximal ideals is generalized. Finally, we characterize relevant properties of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals.</p></abstract>

$ \Omega $-result for the index of composition of an integral ideal

<abstract> Every nonzero integral ideal can be expressed as the product of finite prime ideals in Dedekind domain. For each integral ideal $ \mathfrak{A} $, it is essential to measure the multiplicity of its prime ideal factors. We define $ \lambda(\mathfrak{A}): = \frac{\log N(\mathfrak{A})}{\log \gamma(\mathfrak{A})} $ to be the index of composition of $ \mathfrak{A} $, where $ \gamma(\mathfrak{A}) = \prod_{\mathfrak{P}|\mathfrak{A}}N(\mathfrak{P}) $ and $ N(\mathfrak{A}) $ is the norm of ideal $ \mathfrak{A} $. In this paper, we obtain an $ \Omega $-result for the mean value of the index of composition of integral ideal. </abstract>

Upper bounds on residues of Dedekind zeta functions of non-normal totally real cubic fields

Various bounds on the absolute values of $L$-functions of number fields at $s=1$ and on residues at $s=1$ of Dedekind zeta functions of a number field $\mathbb {L}$ are known. Also, better bounds depending on the splitting behavior of given prime ideals o

Distributive lattices with strong endomorphism kernel property as direct sums

In this paper,for a distributive lattice $mathcal L$, we study and compare some lattice theoretic features of $mathcal L$ and topological properties of the Stone spaces ${rm Spec}(mathcal L)$ and ${rm Max}(mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $Gamma(mathcal L)$.Among other things,we show that the Goldie dimension of $mathcal L$ is equal to the cellularity of the topological space ${rm Spec}(mathcal L)$ which is also equal to the clique number of the zero-divisor graph $Gamma(mathcal L)$. Moreover, the domination number of $Gamma(mathcal L)$ will be compared with the density and the weight of the topological space ${rm Spec}(mathcal L)$. For a $0$-distributive lattice $mathcal L$, we investigate the compressed subgraph $Gamma_E(mathcal L)$ of the zero-divisor graph $Gamma(mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $mathcal L$.

Properties of strongly prime ideals and C-ideals in posets

In this paper, the concepts of C-ideal are defined and explored the various properties C-ideals in posets. The equivalent conditions for an ideal to be a C-ideal is obtained. Further the relations between strongly prime ideals and C-ideals are discussed.

Applying the Cz\'edli-Schmidt sequences to congruence properties of planar semimodular lattices

Following G.~Gratzer and E.~Knapp, 2009, a planar semimodular lattice $L$ is \emph{rectangular}, if~the left boundary chain has exactly one doubly-irreducible element, $c_l$, and the right boundary chain has exactly one doubly-irreducible element, $c_r$, and these elements are complementary. The Czedli-Schmidt Sequences, introduced in 2012, construct rectangular lattices. We use them to prove some structure theorems. In particular, we prove that for a slim (no $\mathsf{M}_3$ sublattice) rectangular lattice~$L$, the congruence lattice $\Con L$ has exactly $\length[c_l,1] + \length[c_r,1]$ dual atoms and a dual atom in $\Con L$ is a congruence with exactly two classes. We also describe the prime ideals in a slim rectangular lattice.

Zariski topology on the spectrum of graded pseudo prime submodules

In this article, we introduce the concept of graded pseudo prime submodules of graded modules that is a generalization of the graded prime ideals over commutative rings. We study the Zariski topology on the graded spectrum of graded pseudo prime submodules. We clarify the relation between the properties of this topological space and the algebraic properties of the graded modules under consideration.

On graded primary-like submodules of graded modules over graded commutative rings

Let G be a group with identity e. Let R be a G-graded commutative ring andM a graded R-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to and extra properties of these graded submodules.

Correction to: Prime ideals and generalized derivations with central values on rings

The article Prime ideals and generalized derivations with central values on rings written by Mamouni Abdellah, Oukhtite Lahcen and Zerra Mohammed was originally published electronically on the publisher’s internet portal (currently Springerlink) on November 20, 2020 with open access.

Catenarity in quantum nilpotent algebras

In this paper, it is established that quantum nilpotent algebras (also known as CGL extensions) are catenary, i.e., all saturated chains of inclusions of prime ideals between any two given prime ideals P ⊊ Q P \subsetneq Q have the same length. This is achieved by proving that the prime spectra of these algebras have normal separation, and then establishing the mild homological conditions necessary to apply a result of Lenagan and the first author. The work also recovers the Tauvel height formula for quantum nilpotent algebras, a result that was first obtained by Lenagan and the authors through a different approach.

The inverse sieve problem for algebraic varieties over global fields

Let K be a global field and let Z be a geometrically irreducible algebraic variety defined over K . We show that if a big set S\subseteq Z of rational points of bounded height occupies few residue classes modulo \mathfrak{p} for many prime ideals \mathfrak{p} , then a positive proportion of S must lie in the zero set of a polynomial of low degree that does not vanish at Z . This generalizes a result of Walsh who studied the case when S\subseteq \{0,\ldots ,N\}^{d} .

Central lifting property for orthomodular lattices

Abstract We introduce and investigate central lifting property (CLP) for orthomodular lattices as a property whereby all central elements can be lifted modulo every p-ideal. It is shown that prime ideals, maximal ideals and finite p-ideals have CLP. Also Boolean algebras, simple chain finite orthomodular lattices, subalgebras of an orthomodular lattices generated by two elements and finite orthomodular lattices have CLP. The main results of the present paper include the investigation of CLP for principal p-ideals and finite direct products of orthomodular lattices.

Structure of spectra of precompletions

Let T be a complete local (Noetherian) ring, and let A be a local subring of T such that the completion of A with respect to its maximal ideal is T. We investigate the possible structures of the partially ordered set Spec(A). Specifically, we explore the minimal prime ideals of A and their formal fibers, the maximal chains of prime ideals in A, and the number of prime ideals in A containing combinations of minimal prime ideals of A.

Relative Ideals in Ordered Semigroups

In this paper, after introducing the notion of relative ideals in ordered semigroups, some important properties of these ideals are studied. Then notions of relatively prime and relatively weakly prime ideals are defined and some vital results have been proved. We also find some conditions under which relative ideals become left, right and two-sided ideals in any regular ordered semigroup. Finally, the notion of maximal and minimal relative ideals are introduced and some of their properties are studied.

Prime ideals and generalized derivations with central values on rings

Our goal in the present paper is to study a connection between the commutativity of rings and the behaviour of its generalized derivations. More specifically, we investigate commutativity of quotient rings R/P where R is any ring and P is a prime ideal of R which admits generalized derivations satisfying certain algebraic identities acting on prime ideal P without the primeness (semi-primeness) assumption on the considered ring. This approach allows us to generalize some well known results characterizing commutativity of rings.

On α-prime and weakly α-prime ideals in semirings

Characterizations of [Formula: see text]-prime ideals, weakly [Formula: see text]-prime ideals in a semiring [Formula: see text] are investigated. If [Formula: see text] is a [Formula: see text]-ideal of a semiring [Formula: see text], then a relation between the set of [Formula: see text]-prime ideals (weakly [Formula: see text]-prime ideals) in [Formula: see text] and the set of [Formula: see text]-prime ideals (weakly [Formula: see text]-prime ideals) in the quotient semiring [Formula: see text] is obtained.

$Gr$-$(2,n)$-ideals in graded commutative rings

Let $G$ be a group with identity $e$ and let $R$ be a $G$-graded ring. In this paper, we introduce and study the concept of graded $(2,n)$-ideals of $R$. A proper graded ideal $I$ of $R$ is called a graded $(2,n)$-ideal of $R$ if whenever $rst\in I$ where $r,s,t\in h(R)$, then either $rt\in I$ or $rs\in Gr(0)$ or $st\in Gr(0)$. We introduce several results concerning $gr$-$(2,n)$-ideals. For example, we give a characterization of graded $(2,n)$-ideals and their homogeneous components. Also, the relations between graded $(2,n)$-ideals and others that already exist, namely, the graded prime ideals, the graded 2-absorbing primary ideals, and the graded $n$-ideals are studied.

Associated primes and integral closure of Noetherian rings

Let A be a Noetherian ring and B be a finitely generated A-algebra. Denote by A' the integral closure of A in B. We give necessary and sufficient conditions for prime ideals to be in Ass_{A}(B/A') and Ass_{A'}(B/A') generalizing and strengthening classical results for rings of special type.