Prime Ideals Research Articles

A characterization of ramification groups via jet algebras

AbstractWe 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 $$R$$ R -algebra and compute their kernels for algebras of finite type over a field with separably generated residue fields.

\(\mathcal{L}\)−weakly 1−Absorbing Prime Ideals and Filters

In this manuscript, we have presented the concept of \(\mathcal{L}\)-weakly 1-absorbing prime ideals and \(\mathcal{L}\)-weakly 1-absorbing prime filters within an ADL. Mainly, we illustrate the connections between \(\mathcal{L}\)-weakly prime ideals (filters) and \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters), as well as between \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters) and \(\mathcal{L}\)-weakly 2-absorbing ideals (filters). Lastly, we have shown that both the image and inverse image of \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters) result in \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters).

Associated varieties of minimal highest weight modules

Let g \mathfrak {g} be a complex simple Lie algebra. A simple g \mathfrak {g} -module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for g \mathfrak {g} . This classification extends the work of Joseph [Ann. Sci. École Norm. Sup. (4) 31 (1998), 17–45], which focused on categorizing minimal highest weight modules annihilated by completely prime ideals. Furthermore, we have determined the associated varieties of these modules. In other words, we have identified all possible weak quantizations of minimal orbital varieties.

On rings of integer-valued rational functions

Let D ⊆ B be an extension of integral domains and E a subset of the quotient field of D. We introduce the ring of D -valued B-rational functions on E, denoted by Int B R ( E , D ) , which naturally extends the concept of integer-valued polynomials, defined as Int B R ( E , D ) : = { f ∈ B ( X ) ; f ( E ) ⊆ D } . The notion of Int B R ( E , D ) boils down to the usual notion of integer-valued rational functions when the subset E is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from Int R ( E , D ) to D.

Controlling Formal Fibers of Countably Many Principal Prime Ideals

Controlling Formal Fibers of Countably Many Principal Prime Ideals

How many principal prime ideals are there in a polynomial ring?

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an indeterminate over [Formula: see text], [Formula: see text] be the polynomial ring over [Formula: see text], [Formula: see text] be the power series ring over [Formula: see text], and [Formula: see text] be a principal prime ideal of [Formula: see text] with [Formula: see text]. It is well known that if [Formula: see text] is an integral domain, then [Formula: see text] is a DVR and [Formula: see text] has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) [Formula: see text] is a field, (ii) [Formula: see text] is a DVR, but (iii) there is a ring [Formula: see text] such that [Formula: see text] has no principal prime ideal. We also study the maximal ideals of [Formula: see text] that are principal.

On the Uniform Dimension and the Associated Primes of Skew PBW Extensions

In this paper, we study the uniform dimension and the associated prime ideals of induced modules over skew PBW extensions.

A generalization of prime ideals in MV-algebras

In this paper, we introduce a special type of prime ideals in [Formula: see text]-algebras. We define the concept of a [Formula: see text]-prime ideal, which is a proper ideal [Formula: see text] that preserves the property of being prime with respect to the ideal [Formula: see text]. If [Formula: see text] is not a subset of [Formula: see text], then we call the [Formula: see text]-prime ideal an [Formula: see text]-prime ideal. We investigate the relationships between these ideals and their connections to minimal prime ideals and maximal ideals. Additionally, we explore various properties such as extension, transitivity, meet, join, and annihilator. Finally, we study prime ideals in [Formula: see text]-algebras of continuous functions.

Graded pseudo weakly prime spectrum of graded topological modules

In this study, we introduce graded pseudo weakly prime submodules of G-graded R-modules, which are an extension of graded weakly prime ideals over G-graded rings. On the graded spectrum of graded pseudo weakly prime submodules, we investigate the Zariski topology. Different aspects of this topological space are investigated, and they are linked to the algebraic properties of the G-graded R-modules under study.

On Energy of Prime Ideal Graph of a Commutative Ring Associated with Transmission-Based Matrices

This research explores the energy of the prime ideal graph of a commutative ring. The study demonstrates the energy formula of the graph associated with transmission-based matrices. Through research, the findings highlight the distance, Wiener-Hosoya, and distance signless Laplacian matrices. It should be noted that the distance and Wiener-Hosoya energies are always twice their spectral radius, meanwhile, it does not hold for distance signless Laplacian energy.

Attached prime ideals over skew Ore polynomials

In this paper, we investigate the attached prime ideals of inverse polynomial modules over skew Ore polynomials.

Equivariant prime ideals for infinite dimensional supergroups

Let A A be a commutative algebra equipped with an action of a group G G . The so-called G G -primes of A A are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When G G is an infinite dimensional group, these ideals can be very subtle: for instance, distinct G G -primes can have the same radical. In previous work, the second author showed that if G = G L ∞ G=\mathbf {GL}_{\infty } and A A is a polynomial representation, then these pathologies disappear when G G is replaced with the supergroup G L ∞ | ∞ \mathbf {GL}_{\infty |\infty } and A A with a corresponding algebra; this leads to a geometric description of G G -primes of A A . In the present paper, we construct an abstract framework around this result, and apply the framework to prove analogous results for other (super)groups. We give some applications to the isomeric determinantal ideals (commonly known as “queer determinantal ideals”).

Graded Weakly Strongly Quasi-Primary Ideals over Commutative Graded Rings

In this article, we introduce and examine the concept of graded weakly strongly quasi primary ideals. A proper graded ideal P of R is said to be a graded weakly strongly quasi primary (shortly, Gwsq-primary) ideal if whenever 0≠xy∈P, for some homogeneous elements x,y∈R, then x2∈P or yn∈P, for some positive integer n. Many examples and properties of Gwsq-primary ideals are given. Among several results, we compare Gwsq-primary ideals and other classical graded ideals such as graded strongly quasi primary ideals, graded weakly primary ideals and graded weakly 2-prime ideals etc.

On local divisor class groups of complete intersections

Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let A be a local Cohen–Macaulay ring.(1)A is a normal domain if and only if A is a normal domain in codimension at most 1.(2)Suppose that A is a normal domain and a complete intersection. Then the divisor class group of A is a subgroup of the projective limit of the divisor class groups of the localizations Ap, where p runs through all prime ideals of height at most 3 in A. We use this fact to describe for an integral Noetherian locally complete intersection scheme X the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if X is locally a UFD.

Hurwitz series rings satisfying a zero divisor property

In this paper, we study zero divisors in the Hurwitz series rings and the Hurwitz polynomial rings over general non-commutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (the Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of the Hurwitz series type (so reduced) ring [Formula: see text] with [Formula: see text] on annihilator ideals, HR (the Hurwitz series ring with coefficients over [Formula: see text]) has finitely many minimal prime ideals, say [Formula: see text] such that [Formula: see text] and [Formula: see text] for some minimal prime ideal [Formula: see text] of [Formula: see text] for all [Formula: see text], where [Formula: see text] are all minimal prime ideals of [Formula: see text]. Additionally, we construct various types of (the Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.

Maximal Ideal and Prime Ideal in an $L$-Lattice

Maximal Ideal and Prime Ideal in an $L$-Lattice

Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice

Let ℜ be the finite commutative ring with unity and <i>I<sub>s</sub></i> be the S-prime ideal of a ring ℜ. The set <i>L<sub>s</sub></i> forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (<i>L<sub>s</sub></i>, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

Monotone-Cevian and Finitely Separable Lattices

AbstractA distributive lattice with zero is completely normal if its prime ideals form a root system under set inclusion. Every such lattice admits a binary operation $$(x,y)\mapsto x\mathbin {\smallsetminus }y$$ ( x , y ) ↦ x \ y satisfying the rules $$x\le y\vee (x\mathbin {\smallsetminus }y)$$ x ≤ y ∨ ( x \ y ) and $$(x\mathbin {\smallsetminus }y)\wedge (y\mathbin {\smallsetminus }x)=0$$ ( x \ y ) ∧ ( y \ x ) = 0 — in short a deviation. In this paper we study the following additional properties of deviations: monotone (i.e., isotone in x and antitone in y) and Cevian (i.e., $$x\mathbin {\smallsetminus }z\le (x\mathbin {\smallsetminus }y)\vee (y\mathbin {\smallsetminus }z)$$ x \ z ≤ ( x \ y ) ∨ ( y \ z ) ). We relate those matters to finite separability as defined by Freese and Nation. We prove that every finitely separable completely normal lattice has a monotone deviation. We pay special attention to lattices of principal $$\ell $$ ℓ -ideals of Abelian $$\ell $$ ℓ -groups (which are always completely normal). We prove that for free Abelian $$\ell $$ ℓ -groups (and also free $$\Bbbk $$ k -vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean $$\ell $$ ℓ -group with strong unit, of cardinality $$\aleph _1$$ ℵ 1 , whose principal $$\ell $$ ℓ -ideal lattice does not have a monotone deviation.

Exploring Commutativity via Generalized (α, β)-Derivations Involving Prime Ideals

Exploring Commutativity via Generalized (α, β)-Derivations Involving Prime Ideals