Shakir Ali

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Verified

Shakir verified their affiliation via an institutional email.

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• Ph.D.
• Professor (Full) at Aligarh Muslim University

Let R be an associative ring with an involution ’∗’. In this article, we introduce the notions of centrally-extended generalized Jordan ∗-derivation, centrally extended Jordan left ∗-centralizer and characterize these mappings in involutive prime rings.

In this paper, we explore and examine a new class of maps known as reverse homoderivations. A reverse homoderivation refers to an additive map g defined on a ring T that satisfies the condition, g(ϑℓ)=g(ℓ)g(ϑ)+g(ℓ)ϑ+ℓg(ϑ), for all ϑ,ℓ∈T. We present various results that enhance our understanding of reverse homoderivations, including their existence...

In this paper, we monitor the behavior of a prime ring with involution of second kind. Such rings are studied through the prism of higher derivations satisfying certain differential identities. Precisely, we prove that for a higher derivation D = ( d i ) i ∈ ℕ ∪ { 0 } {D=(d_{i})_{i\in\mathbb{N}\cup\{{0}\}}} , if we are able to establish the identit...

The present paper aims to prove the invariance of minimal prime ideals under higher derivations. Later on, with the help of invariance property of minimal prime ideals under higher derivations, we establish the *-version of Posner's second theorem for higher derivations in semiprime rings with involution ' * '.

Let m ≥ 1 {m\geq 1} , n ≥ 3 {n\geq 3} be fixed integers, ℛ {\mathscr{R}} be a unital ring with a nontrivial idempotent, and let ζ m : ℛ → ℛ {\zeta_{m}:\mathscr{R}\to\mathscr{R}} be a multiplicative Jordan n -higher derivation ( n ≥ 3 ) {(n\geq 3)} . In this paper, we prove that if c ⁢ h ⁢ a ⁢ r ⁢ ( ℛ ) ≠ 2 ⁢ 𝑎𝑛𝑑 ≠ n - 1 {char(\mathscr{R})\neq 2~{}\...

This paper focuses on examining a new type of n-additive maps called the symmetric reverse n-derivations. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their...

This paper focuses on examining a new type of $n$-additive maps called the symmetric reverse $n$-derivations. As implied by its name, it combines the ideas of $n$-additive maps and reverse derivations, with a $1$-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particular...

This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring $\mathfrak{S}$, which is defined as $\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v...

Sharing confidential information is a critical concern in today’s world. Secret sharing schemes facilitate the sharing of secrets in a way that ensures only authorized participants (shareholders) can access the secret using their allocated shares. Hierarchical secret sharing schemes (HSSSs) build upon Shamir’s scheme by organizing participants into...

Topological indices are numerical parameters that indicate the topology of graphs or hypergraphs. A hypergraph H=(V(H),E(H)) consists of a vertex set V(H) and an edge set E(H), where each edge e∈E(H) is a subset of V(H) with at least two elements. In this paper, our main aim is to introduce a general hypergraph structure for the prime ideal sum (PI...

Let m , n {m,n} be the fixed positive integers and let ℛ {\mathcal{R}} be a ring. In 1978, Herstein proved that a 2-torsion free prime ring ℛ {\mathcal{R}} is commutative if there is a nonzero derivation d of R such that [ d ⁢ ( ϱ ) , d ⁢ ( ξ ) ] = 0 {[d(\varrho),d(\xi)]=0} for all ϱ , ξ ∈ R {\varrho,\xi\in R} . In this article, we study the above...

Let $N$ be a left near-ring and let $\sigma, \tau$ be automorphisms of $N$. An additive mapping $d : N \longrightarrow N$ is called a $(\sigma, \tau)$-derivation on $N$ if $d(xy) = \sigma (x)d(y) + d(x)\tau (y)$~for all $x,y \in N$. In this paper, we obtain Leibniz' formula for $(\sigma, \tau)$-derivations on near-rings which facilitates the proof...

Sharing confidential information is a critical concern in today’s world. Secret sharing schemes facilitate the sharing of secrets in a way that ensures only authorized participants (shareholders) can access the secret using their allocated shares. Hierarchical secret sharing schemes (HSSSs) build upon Shamir’s scheme by organizing participants into...

Let n≥2 be a fixed integer and A be a C∗-algebra. A permuting n-linear map G:An→A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D:An→A such that Gς1,ς2,…,ςiςi′,…,ςn=Gς1,ς2,…,ςi,…,ςnςi′+ςiD(ς1,ς2,…,ςi′,…,ςn) holds ∀ςi,ςi′∈A. In this paper, we investigate the structure of C∗-algebras involving generalized...

Let k>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>1$$\end{document} be a fixed integer. In Gupta and Ray (Cryptography and Communications 7: 257–287, 2015), prov...

Let $n \geq 2$ be a fixed integer and $\mathcal{A}$ be a $C^*$-algebra. A permuting $ n $-linear map $ \mathcal{G} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ is known to be symmetric generalized $n$-derivation if there exists a symmetric $n$-derivation $ \mathfrak{D} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ such that $ \mathcal{G} \left(x_{1}...

Background: Glycine is a conditional non-essential amino acid in human and other mammals. It is abundant in the liver and is known for a wide spectrum of characteristics including the antioxidant, antiinflammatory, immunomodulatory, and cryoprotective effects. The amino acid is a naturally occurring osmolyte compatible with protein surface interact...

The present study examines the impact of permissive parenting style, authoritarian parenting style, and authoritative parenting style on aggression and emotional intelligence among college and university students of Gilgit Pakistan. The participant of the study consists of (n=222) students of both genders within the age range of 15 to 30 years. The...

Cyclic codes over non-chain ring R(α 1 , α 2 ,. .. , α s) and their applications to quantum and DNA codes Abstract: Let s ≥ 1 be a fixed integer. In this paper, we focus on generating cyclic codes over the ring R(α 1 , α 2 ,. .. , α s), where α i ∈ F q \{0}, 1 ≤ i ≤ s, by using the Gray map that is defined by the idempotents. Moreover, we describe...

The researchers showed that there is a problem in the dearth of the scales for motor intelligence for elementary students, which requires solving it by designing a measurement tool for it and standardizing it, I followed the descriptive approach using the survey method for a sample totaling (176) students, and the field procedures included taking e...

Let n and m be fixed positive integers. In this paper, we establish some structural properties of prime rings equipped with higher derivations. Motivated by the works of Herstein and Bell-Daif, we characterize rings with higher derivations D = d i i ∈ N satisfying (i) d n x , d m y ∈ Z R for all x , y ∈ R and (ii) d n x , y ∈ Z R for all x , y ∈ R...

An additive mappings δ on R is called a derivation if δ(ab) = δ(a)b + aδ(b) for all a,b ∈ R. If δ is a derivation on a ring R, M and N are right R-modules and f is a right R-linear mapping from M to N, then an additive mapping d : M → N is called a (δ,f)-derivation if d(xa) = d(x)a + f(x)δ(a) for all x ∈ M and a ∈ R. If δ is determined, then the (δ...

This paper deals with some identities on Banach $ ^* $-algebras that are equipped with linear generalized derivations. As an application of one of our results, we describe the structure of the underlying algebras. Precisely, we prove that for a linear generalized derivation $ F $ on a Banach $ ^* $-algebra $ A $, either we obtain the existence of a...

Let n≥1 be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of Zn, where n=pα,p2q,p2q2,pqr,p3q,p2qr, and pqrs for the different prime integers p,q,r, and s. Moreover, we construct M-polynomials and CoM-polynomials using...

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I 1 , and L m : A → A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type...

Let $m$ and $n$ be the fixed positive integers. Suppose $\mathcal{A}$ is a von Neuman algebra with no central summands of type $I_{1}$ and $L_{m}$ be a Lie type higher derivation i.e., an additive (linear) map $L_{m} :\mathcal{A}\to \mathcal{A}$ such that \begin{equation}\label{def2} \[L_{m}(p_{n}(\mathfrak{S}_{1},\mathfrak{S}_{2},\cdots,\mathfrak{...

The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations and describe the structure of the quotient ring S/P, where S is any ring and P is a prime ideal of S. The symmetricity of additive mappings allows us to...

Let 𝒜 be an ∗-algebra and k ∈ ℤ+. For A,B ∈𝒜, the product defined by ∗[A,δA (B)λ∗] k =∗[A,∗[A,δA(B)λ∗] k−1]1, where ∗[A,δA(B)λ∗] 0 = δA(B)λ∗ and ∗[A,δA(B)λ∗] 1 = AδA(B)λ∗− δ A(B)λ∗A∗, is called λ-k-skew Lie product of A and B, where δA(B)λ∗ = AB − λBA∗. This paper discusses some useful characteristics of λ-k-skew Lie products on prime ∗-algebras. M...

Let R be a ring and let n ≥ 1 {n\geq 1} be a fixed integer. An additive mapping h of a ring R into itself is called an n -Jordan homoderivation if h ⁢ ( x n ) = ( h ⁢ ( x ) + x ) n - x n {h(x^{n})=(h(x)+x)^{n}-x^{n}} holds for all x ∈ R {x\in R} . In this paper, we initiate the study of n -Jordan homoderivations on rings. Precisely, we characterize...

Throughout this paper, we will establish a comprehensive theoretical foundation and rigorously develop the methodology to investigate the structure of quotient ring S/P under the influence of symmetric generalized n-derivations, where S represents an arbitrary ring, and P signifies a prime ideal of S satisfying certain algebraic identities acting o...

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]〈v3−α22v〉 and Q=Fq[u,v]〈u2−α12,v3−α22v〉 are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum...

Codes over FQ PQ with Applications Entropy 2023, 25, 1161

A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several...

A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation is central, then the ring is commutative. In the present paper, we extend this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several oth...

Background: This paper aims to investigate the strength of the relationship between Emotional Intelligence and Life Satisfaction with Resilience among college students living in Turbat, Balochistan, Pakistan. Methodology: Inform consent was taken from the participants before using a purposive sampling technique. They were categorized into two group...

Salinity is an imbalanced concentration of mineral salts in the soil or water that causes yield loss in salt-sensitive crops. Rice plant is vulnerable to soil salinity stress at seedling and reproductive stages. Different non-coding RNAs (ncRNAs) post-transcriptionally regulate different sets of genes during different developmental stages under var...

The present paper aims to investigate the containment of nonzero central ideal in a ring $ \mathcal{R} $ when the trace of symmetric $ n $-derivations satisfies some differential identities. Lastly, we prove that in a prime ring $ \mathcal{R} $ of suitable torsion restriction, if $ \mathfrak{D}, \mathcal{G} : \mathcal{R}^n \rightarrow \mathcal{R} $...

Let $ \mathcal{A} $ be a Banach algebra and $ n > 1 $, a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if $ \mathcal{A} $ is a prime Banach algebra admitting a continuous projection $ \mathcal{P} $ with image in $ \mathcal{Z}(\mathcal{A}) $ such th...

The main objective of this paper is to study about the action of generalized derivations on prime ideals of an arbitrary ring with involution.

The main goal of this paper is to characterize Jordan two-sided centralizers, Jordan centralizers and related maps on triangular rings without identity. As an application of our main theorem, we characterize Jordan generalized derivations on triangular rings. Precisely, we prove that every Jordan generalized derivation on a triangular ring is a two...

Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/〈u12−α2,u22−β2,u1u2−u2u1〉. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4...

Let R be an associative ring and let s≥1 be a fixed integer. An additive map h on R is called a homoderivation if h(xy)=h(x)h(y)+h(x)y+xh(y) holds for all x,y∈R. In 1978, Herstein proved that a prime ring R of char(R)≠2 is commutative if there is a nonzero derivation d of R such that [d(x), d(y)]=0 for all x, y∈R. The main objective of this paper i...

Let k, m be positive integers and F 2 m be a finite field of order 2 m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring S k = F 2 m [v 1 ,v 2 ,...,v k ] v 2 i −α i v i ,v i v j −v j v i , for i, j = 1, 2, 3,. .. , k, where α i is the non-zero element of F 2 m. As an application,...

Let $ \mathfrak{S} $ be a ring. The main objective of this paper is to analyze the structure of quotient rings, which are represented as $ \mathfrak{S}/\mathfrak{P} $, where $ \mathfrak{S} $ is an arbitrary ring and $ \mathfrak{P} $ is a prime ideal of $ \mathfrak{S} $. The paper aims to establish a link between the structure of these rings and the...

Let $ n \geq 1 $ be a fixed integer. Within this study, we present a novel approach for discovering reversible codes over rings, leveraging the concept of $ r $-glifted polynomials. This technique allows us to achieve optimal reversible codes. As we extend our methodology to the domain of DNA codes, we establish a correspondence between $ 4t $-base...

This paper addresses the outcomes that elucidate the characteristics of symmetric linear n-derivations within the realm of C *-algebras. Basically, we show that in a primitive C *-algebra A, if D, G : A n → A are two nonzero symmetric linear n-derivations such that f(a)a + ag(a) = 0 holds ∀ a ∈ W, a nonzero left ideal of A where f and g are traces...

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jord...

The purpose of this paper is to study the \(*\)-differential identities in prime rings with involution \(*\) which admits a pair of derivations. In particular, if a prime ring with involution \(*\) of the second kind with char\((R)\ne 2\) admits derivations \(d_1\) and \(d_2\) such that $$d_1([x,x^*])+[d_2(x),d_2(x^*)]\pm [x,x^*]\in Z(R)~~\text{ fo...

Let n > 1 be a fixed positive integer and S be a subset of a ring R. A mapping ζ of a ring R into itself is called n-skew-commuting on S if ζ(x)x n + x n ζ(x) = 0, ∀ x ∈ S. The main aim of this paper is to describe n-skew-commuting mappings on appropriate subsets of R. With this, many known results can be either generalized or deduced. In particula...

Citrullination process is a natural pathway which happened at the time of cell dying, but during removal of these products, PAD enzymes may release causing stimuli to the immune system. This can start a consequence of processes and enhancement of cancer disease. In this review, we will focus on this marker and significance of its application for ea...

Let m≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 1$$\end{document} be a fixed integer and q be an odd prime such that q=pm\documentclass[12pt]{minimal} \use...

Let R be a noncommutative prime ring with involution of the second kind and H(R) and S(R) be the set of symmetric and skew symmetric elements of R. The aim of the present paper is to show that every strong commutativity preserving endomorphism on H(R) and S(R) is strong commutativity preserving on R.

Let R be a prime ring, α an automorphism of R and b an element of Q, the maximal right ring of quotients of R. The main purpose of this paper is to characterize skew b-derivations in prime rings which satisfy various differential identities. Further, we provide an example to show that the assumed restrictions cannot be relaxed.

Artificial recharging of the abandoned and/or functional dug wells through a recharge filter is an efficient and economical method for mitigating and maintaining depleting water tables. An optimum sizing of the recharge filter is necessary to avoid its over-design or under-design and has a minimum cost. Optimum sizing of the recharge filter for the...

Let P be a partially ordered set (poset). The objective of the present paper is to introduce and study the idea of symmetric bi-derivations of posets. Several characterization theorems involving symmetric bi-derivations are given.

In the present article, we characterize generalized derivations and left multipliers of prime rings involving commutators with idempotent values. Precisely, we prove that if a prime ring of characteristic different from 2 admits a generalized derivation G with an associative nonzero derivation g of R such that [G(u), u] n = [G(u), u] for all u ∈ {[...

The agro‐ecosystem is one of the world's most important ecosystems, which provides food, fiber, fodder, and fuel, as well as water and climate regulation, and cultural services to humans for their well‐being. These agro‐ecosystem services also provide significant economic benefits to people and the socio‐economic development of a country. However,...

Let R be a *-ring. For any x,y∈R, we denote the skew Lie product of x and y by ▿[x,y]=xy-yx∗. An additive mapping F:R→R is called a generalized derivation if there exists a derivation d such that F(xy)=F(x)y+xd(y) for all x,y∈R. The objective of this paper is to chracterize generalized derivations and to describe the structure of prime rings with i...

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jord...

Let \(n\ge 1\) be a fixed integer and R be a ring with involution ‘\(*\)’. For any two elements x and y in R, the n-skew Lie product and n-skew Jordan product are given by \(\triangledown [x,y]_n=\triangledown [x,\triangledown [x,y]_{n-1}]\) with \(\triangledown [x,y]_0=y, ~\triangledown [x,y]_1= \triangledown [x,y]=xy-yx^* ,~ \triangledown [x,y]_2...

Herstein proved that a prime ring R of char(R)≠2 is commutative if there is a nonzero derivation d of R such that [d(x),d(y)]=0 for all x,y∈R. The aim of this paper is to prove the *-version of Herstein’s result with a pair of derivations on prime ideals of a ring with involution. Precisely, we prove the following result: let R be a ring with invol...

Let R be a *-ring and n ≥ 1 be an integer. The objective of this paper is to introduce the notion of n-skew centralizing maps on *-rings, and investigate the impact of these maps. In particular, we describe the structure of prime rings with involution ′ * ′ such that * [x, d(x)]n ∈ Z(R) for all x ∈ R (for n = 1, 2), where d : R → R is a nonzero der...

Let α be non-zero element of F q , where F q is a field of order q and q is a power of an odd prime p. The main goal of this paper is to study structural properties of cyclic codes over the finite ring R = F q [u 1 , u 2 ]/⟨u 1 2 − α 2 , u 2 2 − 1, u 1 u 2 − u 2 u 1 ⟩. Moreover, as an application, we construct quantum-error-correcting (QEC) codes.

In the present article, we characterize generalized derivations and left multipliers of prime rings involving commutators with idempotent values. Precisely, we prove that if a prime ring of charac- teristic different from 2 admits a generalized derivation G with an associative nonzero derivation g of R such that [G(u); u]n = [G(u); u] for all u 2 f...

ABSTRACT Objectives: This study aims to analyze epidemiological data that diabetes leads to increased susceptibility to initial tuberculosis infection or if diabetes leads to increased progression from latent tuberculosis to active tuberculosis. Methods: A simplified MEDLINE search method has been used in this study. The PubMed’s Clinical Queries...

Constacyclic codes over the ring F p [u, v]/ u 2 − 1, v 3 − v, uv − vu and their applications Abstract The objective of this paper is to investigate the structural properties of (λ 1 +uλ 2 + vλ 3 + v 2 λ 4 + uvλ 5 + uv 2 λ 6)-constacyclic codes over the ring F p [u, v]/ u 2 − 1, v 3 − v, uv − vu for odd prime p. Precisely, we prove that the Gray im...

In recent years, microRNAs (miRNAs) and tRNA-derived RNA fragments (tRFs) have been reported extensively following different approaches of identification and analysis. Comprehensively analyzing the present approaches to overcome the existing variations, we developed a benchmarking methodology each for the identification of miRNAs and tRFs, termed a...

Let $R$ be a prime ring and $\alpha,\beta$ be the automorphisms of $R.$ The main aim of this article is to investigates several algebraic identities involving generalized $(\alpha,\beta) -$ derivations acting on Lie ideals of prime rings. More precisely, we study the following identities: (i) $F([x,y]) = \alpha(x)\circ F(y),$ (ii) $F(x\circ y) = [\...

As a reaction to the global climate change, the world is moving towards emission-free transportation by adapting electric vehicles. As the biggest ratio of the transportation is privately owned and operated, its electrification represents one of the biggest challenges for distribution system operators. The contribution provides the basis for the in...

Low-voltage grids have to be planned in order to avoid limit violations at any possible operating point. For this reason, this paper analyses the influence of the two operating points “peak generation” and “peak load” on grid planning considering new emerging trends. The mentioned operating points correspond to distributed generation such as photov...

Degradation of the land ecosystem is a major problem in India due to biotic and abiotic interferences. About 121 M ha land has been degraded and largely falls under rainfed regions. It has negative impacts on agricultural production and economy and the natural environment. In India, the per capita availability of both land and water is declining ex...

Let P be a poset and a : P ! P be a function. The aim of this paper is to introduce and study the notion of skew derivations on P. We prove some fundamental properties of posets involving skew derivations. In particular, apart from proving the other results, we prove that if d and g are two skew derivations of P associated with an automorphism a su...

The main purpose of the study is to investigate the causes of anxiety which effect among the students of English language at secondary level. The data has been collected from the 100 students of secondary level of public schools using a survey questionnaire comprised of 24 items. There are 18 towns in Karachi for this study; as a sample frame New t...

Let $R$ be a ring. An additive mapping $F:R\to R$ is called a generalized derivation if there exists a derivation $d$ of $R$ such that $F(xy)=F(x)y+xd(y)$ for all $ x,y \in R$. The main purpose of this paper is to characterize some specific classes of generalized derivations of rings. Precisely, we describe the structure of generalized derivations...

The Department of Mathematics, Faculty of Science, AMU, Aligarh, has a very strong and active group of researchers working in the core area of pure & applied mathematics, in particular, abstract algebras, analysis and its related topics. Based on the research performance of faculty members and research scholars, the Department of Mathematics has be...

The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m -potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.

Let R be a ring with involution *, skew Lie product of x, y ∈ R is defined by ∇[x, y] = xy − yx *. In this paper, we discuss the commutativity of prime rings with involution equipped with skew Lie product which satisfies the certain *-differential identities involving generalized derivations.

Let R be a noncommutative prime ring with involution 0 à 0 and let Q ms ðRÞ be the maximal symmetric ring of quotients of R: In the present paper, we describe the structure of generalized Jordan Ã-derivations, i.e., additive mappings F : R ! R satisfying Fðx 2 Þ ¼ FðxÞx à þ xdðxÞ for all x 2 R, where d is an associated Jordan Ã-derivation of R: Pre...

The purpose of this paper is to investigate the commutativity of prime rings with involution ‘’ of the second kind in which a pair of derivations satisfy certain -differential identities.

Let {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})} . In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016...

\begin{abstract} Let $\mathcal{R}$ be a prime ring with involution $'*'$ and $\psi: \mathcal{R} \rightarrow \mathcal{R}$ be an endomorphism on $\mathcal{R}$. In this article, we study the action of involution $'*',$ and the effect of endomorphism $\psi$ satisfying $[\psi(x),\psi(x^*)]-[x,x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$....

Let R be a ring with involution and let α and β be endomorphisms of. In this paper we characterise generalised Jordan (,)-higher derivations and related maps on (semi)-prime rings with involution. As consequences of our main theorems, many known results can be either generalised or deduced.

In the present paper, we characterize some additive mappings in prime and semi prime rings with involution. As an application, we describe the structure of Jordan left *-centralizers of semiprime rings with involution.

Our purpose in this paper is to investigate some particular classes of generalized derivations and their relationship with commutativity of prime rings with involution. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide examples to show that the assumed restrictions cannot be relaxed.

The purpose of this paper is to investigate *-differential identities satisfied by pair of derivations on prime rings with involution. In particular, we prove that if a 2-torsion free noncommutative ring R admit nonzero derivations d 1 , d 2 such that [d 1 (x), d 2 (x *)] = 0 for all x ∈ R, then d 1 = λd 2 , where λ ∈ C. Finally, we provide an exam...

Let 1 < k and m, k ∈ Z +. In this manuscript, we analyse the action of (semi)-prime rings satisfying certain differential identities on some suitable subset of rings. To be more specific, we discuss the behaviour of the semiprime ring R satisfying the differential identities ([d([s, t] m), [s, t] m ]) k = [d([s, t] m), [s, t] m ] for every s, t ∈ R...

Let R be a ring. An additive map x → x * of R into itself is called an involution if (i) (xy) * = y * x * and (ii) (x *) * = x hold for all x, y ∈ R. In this paper, we study the effect of involution " * " on prime rings that satisfying certain differential identities. The identities considered in this manuscript are new and interesting. As the appl...

Let R be a ring. An additive mapping \(F : R\rightarrow R\) is called a generalized derivation if there exists a derivation \(d : R\rightarrow R \) such that \( F(x y) = F(x)y + xd(y)\) for all \( x, y \in R\). In this paper, first we describe the structure of prime rings involving automorphisms and then characterized generalized derivations on sem...

Let A be unital prime Banach algebra over ℝ or ℂ with centre and G1, G2 be open subsets of be a continuous linear generalized skew derivation, and be a continuous linear map. We prove that must be commutative if one of the following conditions holds: • For each a ∈ G1, b ∈ G2, there exists an integer m ∈ Z>1 depending on a and b such that either ....

Let ℛ be a ring and Z+ be the set of positive integers. Suppose ξ:ℛ→ℛ is an automorphism of ℛ. In this paper, we study the following functional identity xξxn+xnxξ=0 for every x∈ℛ, and n∈Z+. As an application, we describe the structure of 𝒞*-algebras.