Quotient Ring Research Articles

Identities with b-generalized derivations and generalized skew derivations on prime rings

Let [Formula: see text] be a prime ring of characteristic different from [Formula: see text], [Formula: see text] be the right Martindale quotient ring of [Formula: see text], [Formula: see text] the extended centroid of [Formula: see text], [Formula: see text] be a noncentral Lie ideal of [Formula: see text], [Formula: see text] and [Formula: see text] be two nonzero [Formula: see text]-generalized derivations of [Formula: see text]. Suppose there exist fixed integers [Formula: see text] such that [Formula: see text], for all [Formula: see text], then either [Formula: see text] satisfies the standard identity [Formula: see text] or there is [Formula: see text] such that [Formula: see text], [Formula: see text], for any [Formula: see text], and one of the following holds: (a) there exists [Formula: see text], (b) [Formula: see text] and [Formula: see text] are [Formula: see text]-linearly dependent. Then, in the second part of the paper we prove a similar result in the case [Formula: see text] and [Formula: see text] are generalized skew derivations of [Formula: see text] such that [Formula: see text], for all [Formula: see text].

Zero Divisor Graph of Quotient Ring

Zero Divisor Graph of Quotient Ring

Engel condition of composition of generalized derivations in prime rings

Let [Formula: see text] be a prime ring, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] the extended centroid of [Formula: see text], where [Formula: see text] is the Utumi quotient ring of [Formula: see text]. Suppose that [Formula: see text] are two nonzero generalized derivations of [Formula: see text]. Recently, [Dhara et al., Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48(1) (2020) 154–167] studied the case [Formula: see text] for all [Formula: see text], where [Formula: see text] is a derivation (which is independent of [Formula: see text]) and [Formula: see text] fixed positive integers and obtained the structure of [Formula: see text] and the ring [Formula: see text]. [Liu, An Engel condition with two generalized derivations on Lie ideals of prime rings, J. Algebra and Its Applications, 22(5) (2023) 2350115] studied a similar identity replacing derivation [Formula: see text] with another generalized derivation [Formula: see text]. Recently, [Pandey, Pair of generalized derivations on Lie ideals in prime rings, Asian-European J. Math. 16(1) (2022) 2350002] studied an identity considering composition of mapping, that is [Formula: see text] for all [Formula: see text]. In this paper, we consider the situation containing composition of mappings as well as Engel condition, that is, [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed integers and then obtain the nature of mappings [Formula: see text], [Formula: see text] and the ring [Formula: see text].

B-generalized skew derivations acting on multilinear polynomials in prime rings

Let R be a prime ring of characteristic not equal to 2, Z(R) its center, Qr its right Martindale quotient ring, C its extended centroid, f ( x 1 , … , x n ) a non-central multilinear polynomial over C and u , v ∈ Q r . Suppose that F and H are two b-generalized skew derivations of R with associated terms ( b , α , d ) and ( b , α , δ ) . In this article we study the complete structure of F and H satisfying uF ( f ( X ) ) f ( X ) + F ( f ( X ) ) f ( X ) v = H ( f ( X ) 2 ) for any X = ( r 1 , … , r n ) ∈ R n .

Quotient gamma nearness rings

Quotient gamma nearness rings

Construction of even-level representations of [formula omitted] with residue field of characteristic two

We construct the finite-dimensional continuous complex representations of SL2 over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. We also prove that the complex group algebras of SL2 over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras C[SL2(Z/2rZ)] and C[SL2(F2[t]/(tr))] are not isomorphic for any r≥4.

Characterization of n-Lie-type derivations on triangular rings

By means of the maximal left ring of quotients, this paper proves that every n-Lie m-derivation on a triangular ring can be resolved into a sum of an extremal n-derivation and an n-additive central mapping. As applications, n-Lie m-derivations on some classical triangular rings are characterized.

$\mathscr{T}$-Commuting Generalized Derivations on Ideals and Semi-Prime Ideal

This paper's primary goal is to look at a quotient ring $\mathscr{A}/\mathscr{T}$ structure, where $\mathscr{A}$ is an arbitrary ring and $\mathscr{T}$ is a semi-prime ideal of $\mathscr{A}$. More precisely, we examine the differential identities in a semi-prime ideal of an arbitrary ring involving $\mathscr{T}$-commuting generalized derivation. Furthermore, examples are given to prove that the restrictions imposed on the hypothesisof the various theorems were not superfluous.

A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs

This article investigates the concept of dominant metric dimensions in zero divisor graphs (ZD-graphs) associated with rings. Consider a finite commutative ring with unity, denoted as R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x.y=0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.

Regularity of the edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs.

Reifying dynamical algebra: Maximal ideals in countable rings, constructively

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as “implies 0 = 1”) we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull’s lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.11 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders.

A structure result on modules over polynomial rings

Let [Formula: see text] be a ring. We give the equivalent categories of the module categories over the polynomial ring [Formula: see text] and the quotient ring [Formula: see text]. As applications, we describe explicitly some important classes of modules over [Formula: see text] and [Formula: see text] such as finitely generated modules, projective modules and injective modules.

Composition of generalized derivations acting as a Jordan product in prime rings

Let F, G, and H be generalized derivations with F ≠ 0 on a prime ring R of characteristic different from 2, with Utumi quotient ring U and extended centroid C such that for all r ∈ R n FH ( f ( r ) 2 ) = F ( f ( r ) ) G ( f ( r ) ) + G ( f ( r ) ) F ( f ( r ) ) (1) where f(r) is a multilinear polynomial over C which is noncentral valued on R. In this article we prove that if R does not satisfy the standard identity s 4 then either R satisfies (1) or f ( x 1 , … , x n ) 2 is central valued on R. Also in both the cases we describe all the possible forms of F, G, and H.

On S-valuation domains

Let [Formula: see text] be an integral domain and let [Formula: see text] be a multiplicative subset of [Formula: see text]. In this paper, we study integral domains whose quotient rings are valuation domain. To do this, we introduce the concept of [Formula: see text]-valuation domains. We define [Formula: see text] to be an [Formula: see text]-valuation domain if for each nonzero [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text] divides [Formula: see text] or [Formula: see text] divides [Formula: see text]. Among other things, we show that [Formula: see text] is an [Formula: see text]-valuation domain if and only if [Formula: see text] is a valuation domain. By using this result, we give several valuation-like properties.

Open Covers and Lex Points of Hilbert Schemes Over Quotient Rings via Relative Marked Bases

We introduce the notion of a relative marked basis over quasi-stable ideals, together with constructive methods and a functorial interpretation, developing computational methods for the study of Hilbert schemes over quotients of polynomial rings. Then we focus on two applications. The first has a theoretical flavor and produces an explicit open cover of the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable ideals. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal, against the expectations. The second application has a computational flavor. When the quotient rings are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the Hilbert schemes and find examples of both smooth and singular lex-points.

A fast NTRU software implementation based on 5-way TMVP

The fast and efficient operation of post-quantum cryptographic algorithms has become an important research topic, especially with the recent developments and the process of the PQC Standardization Project by NIST. Various algorithms based on lattices are chosen to be finalists in this project. Number Theoretic Transform (NTT), Toom–Cook, Karatsuba, and Toeplitz matrix–vector product (TMVP) are considered in order to efficiently perform multiplications in quotient polynomial rings required by the cryptographic schemes based on lattices. In this paper, we propose a 5-way split TMVP-based algorithm for multiplication in polynomial quotient rings and determine its computational complexity. The new algorithm is derived from a 5-term polynomial multiplication algorithm using 13 multiplications with coefficients of only 1 or -1 that provide efficiency. We also implement it for all parameter sets of NTRU. The results show that we have up to 34%, 35%, and 157% speed-up against Toom4-Karatsuba implementation in key generation, encapsulation, and decapsulation, respectively.

Pfister forms and a conjecture due to Colliot-Thélène in the mixed characteristic case

Let $R$ be a regular local ring of mixed characteristic $(0,p)$, where $p\neq 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. We fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$ and an invertible element $c$ in $R$. Then the equation $Q(T_{1},\ldots,T_{2^{m}})=c$ has a solution over $R$ if and only if it has a solution over the fraction field $K$.

S‐J‐Ideals: A Study in Commutative and Noncommutative Rings

In this paper, we introduce the concept of S‐‐ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of ‐ideals apply to S‐‐ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m‐system, we define right S‐‐ideals. We demonstrate the equivalence of S‐‐ideals and right S‐‐ideals in commutative rings with identity and provide examples to illustrate the connections between right S‐prime ideals and ‐ideals.

Формы Пфистера и гипотеза Кольe-Телена для случая смешанной характеристики

Let $R$ be a regular local ring of a mixed characteristic $(0,p)$ where $p\neq 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. Fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$ and an invertible element $c$ in $R$. Then the equation $Q(T_{1},\ldots,T_{2^{m}})=c$ has a solution over $R$ if and only if it has a solution over the fraction field $K$.

Polynomial Quotient Rings and Kronecker Substitution for Deriving Combinatorial Identities

Polynomial Quotient Rings and Kronecker Substitution for Deriving Combinatorial Identities