Ideal Class Research Articles

On z-ideals and z-closure operations of semirings, I

The aim of this series of papers is to study z-ideals of semirings. In this article, we introduce some distinguished classes of z-ideals of semirings, which include z-prime, z-semiprime, z-irreducible, and z-strongly irreducible ideals and study some of their properties. Using a z-closure operator, we show the equivalence of these classes of ideals with the corresponding z-ideals that are prime, semirprime, irreducible, and strongly irreducible respectively.

A number‐theoretic classification of toroidal solenoids

AbstractWe classify toroidal solenoids defined by nonsingular ‐matrices with integer coefficients by studying associated first Ĉech cohomology groups. In a previous work, we classified the groups in the case using generalized ideal classes in the splitting field of the characteristic polynomial of . In this paper, we explore the classification problem for an arbitrary .

P-Adic hypergeometric functions and certain weight three newforms

For an odd prime p and a positive integer n, let Gnn[⋯]p denote McCarthy's p-adic hypergeometric function. In this article, we prove p-adic analogue of certain classical hypergeometric identities and using these identities we express the p-th Fourier coefficient of certain weight three newforms in terms of special values of G33[⋯]p. Rodriguez-Villegas conjectured certain supercongruences between values of truncated hypergeometric series and the p-th Fourier coefficients of these newforms. As a consequence of our main results, we obtain another proof of these supercongruences which were earlier proved by Mortenson and Sun.

SIoT identity management with the ability to preserve owner privacy and built‐in resistance to Sybil attacks

SummaryThe Social Internet of Things (SIoT) allows for more efficient and intelligent communication between entities (humans and objects). Managing the digital identity of objects, protecting the owner's privacy, and resisting impersonation attacks are crucial challenges in SIoT. On the other hand, the classic centralized identity provider servers contain valuable information for hackers and malicious parties. Effective solutions can be presented using decentralized technologies such as blockchains. This paper proposes a decentralized identity management system for SIoT based on smart contract technology. This system drives object identity control; conversely, a zero‐knowledge‐based protocol and double‐spend prevention ideas are proposed for privacy preservation and Sybil attack resistance, respectively. The central idea is to implement self‐sovereign identity (SSI) so that even certifiers are not privy to object‐sensitive information. Additionally, it attempts to prevent the owner from creating Sybil objects using the web of trust method. The method's scalability and convergence can be proven by considering all acknowledgment nodes and different paths. This method was compared with the most novel available methods; the results from this comparison depict the scalability and effectiveness of the proposed method for large networks.

A note on repunit number sequence

In this paper, we investigate the classical identities of the repunit sequence with integer indices in light of the properties of Horadan-type sequences. We highlight particularly the Tagiuri-Vajda Identity and Gelin-Cesàro Identity. Additionally, we prove that no repunit is a perfect power, either even or odd. Finally, we address a divisibility criterion for the terms of repunit rn by a prime p and its powers.

Values of p-adic hypergeometric functions and p-adic analogue of Kummer’s linear identity

Let p be an odd prime and [Formula: see text] be the finite field with p elements. This paper focuses on the study of the values of a generic family of hypergeometric functions in the p-adic setting, which we denote by [Formula: see text], where [Formula: see text] and [Formula: see text]. These values are expressed in terms of numbers of zeros of certain polynomials over [Formula: see text]. These results lead to certain p-adic analogues of classical hypergeometric identities. Namely, we obtain p-adic analogues of particular cases of a Gauss’s theorem and a Kummer’s theorem. Moreover, we examine the zeros of these functions. For example, if n is odd, we characterize t for which [Formula: see text] has zeros. In contrast, we show that if n is even, then the function [Formula: see text] has no zeros for any prime p apart from the trivial case when [Formula: see text].

Ideal class groups of division fields of elliptic curves and everywhere unramified rational points

Let E be an elliptic curve over Q, p an odd prime number and n a positive integer. In this article, we investigate the ideal class group Cl(Q(E[pn])) of the pn-division field Q(E[pn]) of E. We introduce a certain subgroup E(Q)ur,pn of E(Q) and study the p-adic valuation of the class number #Cl(Q(E[pn])).In addition, when n=1, we further study Cl(Q(E[p])) as a Gal(Q(E[p])/Q)-module. More precisely, we study the semi-simplification (Cl(Q(E[p]))⊗Zp)ss of Cl(Q(E[p]))⊗Zp as a Zp[Gal(Q(E[p])/Q)]-module. We obtain a lower bound of the multiplicity of the E[p]-component in the semi-simplification when E[p] is an irreducible Gal(Q(E[p])/Q)-module.

Cohen-Macaulay type of orders, generators and ideal classes

In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in étale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of minimal generating sets of its fractional ideals, generalizing known results for Gorenstein and Bass orders. Finally, we give a classification of the ideal classes with multiplicator ring of type 2, with applications to the computations of the conjugacy classes of integral matrices and the isomorphism classes of abelian varieties over finite fields.

Large Pólya groups in simplest cubic fields and consecutive bi-quadratic fields

The Pólya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of consecutiveness of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large Pólya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields.

Quadratic forms and Genus Theory: A link with 2‐descent and an application to nontrivial specializations of ideal classes

AbstractGenus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well‐known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any principal ideal domain (PID) . When , we show that the Genus Theory map is the quadratic form version of the 2‐descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of nontrivial specializations has density 1.

On S-2-Prime Ideals of Commutative Rings

Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expanding this concept into more generalized forms. Among these, S-prime ideals and 2-prime ideals have reaped attention recently. This paper aims to characterize S-2-prime ideals, which serve as a generalization encompassing both 2-prime ideals and S-prime ideals. To accomplish this objective, we construct an ideal which distinct from a multiplicatively closed subset with the help of commutative rings. We investigate the localization and the S-2-prime avoidance lemma in commutative rings. Furthermore, we explore the properties of this class of ideals in trivial ring extensions and amalgamated algebras along an ideal. We delve into S-properties for compactly packedness, compactly 2-packedness and coprimely packedness in trivial ring extentions. Moreover, this notion of ideals helps us to indicate that many results stated in S-prime ideals and 2-prime ideals can be readily expanded to the framework of S-2-prime ideals. Supporting examples also highlight a significant distinction between S-2-prime ideals and stated ideals.

On functional prime ideals in commutative rings

In this paper we introduce the notion of functional prime ideals in a commutative ring. For a (left) R-module M and a functional ϕ (i.e., an R-linear map ϕ from M to R), an ideal I of R is said to be a ϕ -prime ideal if whenever a ∈ R and m ∈ M such that a ϕ ( m ) ∈ I , then a ∈ I or ϕ ( m ) ∈ I . This notion shows its ability to characterize different classes of ideals in terms of functional primeness with respect to specific R-modules. For instance, if the module M is the ideal I itself, then I is ϕ -prime for every ϕ ∈ Hom R ( I , R ) if and only if I is a trace ideal, and if the module M is the dual of I, then I is ϕ -prime for every ϕ ∈ Hom R ( I − 1 , R ) if and only if I is a prime ideal of R, or I is a strongly divisorial ideal. Several results are obtained and examples to illustrate the aims and scopes are provided.

Inequalities for totally nonnegative matrices: Gantmacher–Krein, Karlin, and Laplace

A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is referred to as multiplicative when it compares two collections of products of minors and additive otherwise. Set theoretic operations preserving the class of TN matrices naturally translate into operations preserving determinantal inequalities in this class. We introduce index-row (and index-column) operations that act directly on all determinantal inequalities for TN matrices, and yield further inequalities for these matrices. These operations assist in revealing novel additive inequalities for TN matrices embedded in the classical identities due to Laplace [Mem. Acad. Sciences Paris 1772] and Karlin (1968). In particular, for any square TN matrix A, these derived inequalities generalize – to every ith row of A and jth column of adjA – the classical Gantmacher–Krein fluctuating inequalities (1941) for i=j=1. Furthermore, our index-row/column operations reveal additional undiscovered fluctuating inequalities for TN matrices.The introduced index-row/column operations naturally birth an algorithm that can detect certain determinantal expressions that do not form an inequality for TN matrices. However, the algorithm completely characterizes the multiplicative inequalities comparing products of pairs of minors. Moreover, the underlying index-row/column operations add that these inequalities are offshoots of certain “complementary/higher” ones. These novel results seem very natural, and in addition thoroughly describe and enrich the classification of these multiplicative inequalities due to Fallat–Gekhtman–Johnson [Adv. Appl. Math. 2003] and later Skandera [J. Algebraic Comb. 2004] (and revisited by Rhoades–Skandera [Ann. Comb. 2005, J. Algebra 2006]).

Ring elements of stable range one

A ring element a∈R is said to be of right stable range one if, for any t∈R, aR+tR=R implies that a+tb is a unit in R for some b∈R. Similarly, a∈R is said to be of left stable range one if Ra+Rt=R implies that a+b′t is a unit in R for some b′∈R. In the last two decades, it has often been speculated that these two notions are actually the same for any a∈R. In §3 of this paper, we will prove that this is indeed the case. The key to the proof of this new symmetry result is a certain “Super Jacobson's Lemma”, which generalizes Jacobson's classical lemma stating that, for any a,b∈R, 1−ab is a unit in R iff so is 1−ba. Our proof for the symmetry result above has led to a new generalization of a classical determinantal identity of Sylvester, which will be published separately in [45]. In §§4-5, a detailed study is offered for stable range one ring elements that are unit-regular or nilpotent, while §6 examines the behavior of stable range one elements via their classical Peirce decompositions. The paper ends with a more concrete §7 on integral matrices of stable range one, followed by a final §8 with a few open questions.

The isomorphism problem for ideal class monoids of numerical semigroups

AbstractFrom any poset isomorphic to the poset of gaps of a numerical semigroup S with the order induced by S, one can recover S. As an application, we prove that two different numerical semigroups cannot have isomorphic posets (with respect to set inclusion) of ideals whose minimum is zero. We also show that given two numerical semigroups S and T, if their ideal class monoids are isomorphic, then S must be equal to T.

Arakelov class groups of random number fields

AbstractThe main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen–Lenstra–Martinet heuristic on ideal class groups. To that end, we show that Chinburg’s $$\Omega (3)$$ Ω ( 3 ) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups. As a consequence, we construct a new infinite series of counterexamples to the Cohen–Lenstra–Martinet heuristic, which have the novel feature that their Galois groups are non-abelian.

Short proofs of ideal membership

A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting.We show that the problem of computing cofactor representations with a bounded number of terms is decidable and NP-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gröbner basis algorithms. We show that, for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.

On state monadic MV-algebras

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. In this paper, we extend monadic MV-algebras with a state operator that describes algebraic properties of states. The resulting variety of algebras will be called state monadic MV-algebras. First, we introduce state monadic MV-algebras and establish a natural equivalence between the category of state monadic MV-algebras and the category of state monadic ℓ-groups with strong units. Moreover, we prove that the class of all state monadic ideals in state monadic MV-algebras is a complete Heyting algebra. In particular, by studying extended state monadic ideals, we prove that the set of all stable state monadic ideals in state monadic MV-algebras is a complete Heyting algebra. Also, the class of all involutory state monadic ideals in state monadic MV-algebras is a complete Boolean algebra. Finally, we introduce and characterize some members in the variety of state monadic MV-algebras, which are subdirectly irreducible, simple, semisimple, local and semilocal, respectively.

A Geometric Study of Circle Packings and Ideal Class Groups

A Geometric Study of Circle Packings and Ideal Class Groups

Domestic violence of representatives of the LGBT and queer communities

The article is devoted to the analysis of the problem of domestic violence in the lives of persons with a special sexual identity. The article is based on the study of expert and analytical reports, international monitoring and positions of foreign scientists of various socio-humanitarian and legal schools.
 It is indicated that violence against LGBT people can be considered as a hidden disharmony in societal relations, where the discovery of sexual orientation or gender identity causes a negative resonance, the problem encapsulates a deeper feeling of non­recognition and attacks the basic principles of humanism and equality. Violence directed against LGBT individuals can serve as an indicator of systemic inequality and a crisis of values in society, LGBT violence is a form of assassination of the very essence of dignity and freedom inherent in every person. It becomes a sign of internal dissonance, where the conflict between the individual and society merges into a painful existential drama.
 It is motivated that adult children are less likely to be subjected to domestic violence than LGBT representatives, which is special in the context of socio-cultural, religious and stereotypical ideas that exist in some societies. Parents and relatives often pressure their children about their sexual or gender identity, which can lead to a variety of negative consequences, including emotional and physical abuse. The fear of discovering one's sexual orientation or gender identity becomes a source of conflicts in the family. Parents can react to this with pressure, contempt or refusal to accept, as stereotypes and prejudices can also provoke the formation of a negative attitude of parents from others and the community.
 The peculiarities of the manifestation of domestic violence among representatives of the LGBT and queer communities are determined, in particular, the phenomenon is more widespread compared to people who have a classic sexual identity; issues of domestic violence from parents and family due to sexual identity, often involving coercion into marriage, procreation and confidential therapy; exposed to additional forms of violence (threat of publicizing identity features, homophobia, unprotected relationships).