Mohammad Ashraf

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

Let $\mathfrak{R}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$, and $\textbf{S}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}+v\mathbb{Z}_{2}+uv\mathbb{Z}_{2}$, where $u^{2}=v^{2}=0$, $uv=vu$. In this article, we study $\mathfrak{R}\textbf{S}$-additive cyclic, additive constacyclic, and additive dual codes. We find the structural properties of these codes. The...

Let $\mathcal{A}$ be a unital $\ast$-algebra with a nontrivial projection. In this chapter, it is proved that under some suitable assumptions every nonlinear generalized bi-skew Lie $n$-derivation $\mathcal{G}:\mathcal{A}\to\mathcal{A}$ is of the form $\mathcal{G}(\mathcal{K})=\mathcal{V} \mathcal{K}+\mathcal{D}(\mathcal{K})$ for all $\mathcal{K}\i...

Let $\mathbb{Z}_{2}=\{0,1\}$, $\mathfrak{R_{1}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$ and $\mathfrak{R_{u^{k}}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}+\cdots+u^{k-1 }\mathbb{Z}_{2}$, where $u^{k}=0$. In this article, we study $\mathbb{Z}_{2}\mathfrak{R_{1}}\mathfrak{R_{u^k}}$-additive cyclic, additive dual codes and their structural properties. Th...

Let $${\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where $${\mathbb {Z}}_{4}$$ Z 4 is a ring of integers modulo 4. This ring $${\mathfrak {R}}$$ R is a local non-chain ring of characteristic 4. The main objective of this article is to construct rev...

Let [Formula: see text] be primitive ring of characteristic different from two with nontrivial idempotents. Then every Lie-type derivation can be written in the standard form under certain assumptions.

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...

Let S = ℤ p [ u, v ] /〈u ² , v ² , uv − uv〉 be a semi-local ring, where p is a prime number. In the present article, we determine the generating sets of S and use them to construct the structures of ℤ p S -additive cyclic and constacyclic codes. The minimal polynomials and spanning sets of ℤ p S -additive cyclic and constacyclic codes are also dete...

Let $ \mathcal {R}_{k}= \mathbb {Z}_4[u_{1},u_{2},\ldots,u_{k}]/\langle u_{i}^2-u_{i},u_{i}u_{j}-u_{j}u_{i}\rangle $ be a non-chain ring of characteristic 4, where $ 1\leq i,j\leq k $ and $ k\geq 1 $ . In this article, we discuss reversible cyclic codes of odd lengths over the ring $ \mathcal {R}_{k} $ . We construct bijections between the elements...

Let $\mathfrak{A}=\mathbb Z_4+u\mathbb Z_4+v\mathbb Z_4$, where $u^2=u$, $v^2=v$ and $uv=vu=0,$ be a ring which is an extension of $\mathbb{Z}_{4}$. In this article, we study the structure of cyclic codes over $\mathfrak{A}$ and define a $\mathbb Z_2$-linear isometry $\Phi$ from $\mathfrak{A}^{n}$ to $\mathbb Z^{6n}_2.$ Based on the classical cycli...

Let $\mathfrak{A}$ be a unital prime $\ast$-algebra (over the complex field $\mathbb{C}$) with a nontrivial projection. For any $S_1, S_2,\ldots, S_n \in\mathfrak{A},$ define $q_1(S_1)=S_1, q_2(S_1, S_2)=[S_1, S_2]_\diamond=S_1S_2-S_2S_1^\ast$ and $q_n(S_1, S_2, \ldots, S_n)=[q_{n-1}(S_1, S_2, \ldots, S_{n-1}), S_n]_\diamond$ for all integers $n\ge...

Let $\mathbb{Z}_{2}=\{0,1\}$, $\mathfrak{R_{1}}=\mathbb{Z}_{2}+u\mathbb{Z}_{2}$, where $u^2=0$ and $\mathfrak{R_{2}}{=}\mathbb{Z}_{2}+u\mathbb{Z}_{2}+v\mathbb{Z}_{2}$, where $u^{2}{=}v^{2}=0{=}uv{=}vu$. In this article, we study $\mathbb{Z}_{2}\mathfrak{R_{1}}\mathfrak{R_{2}}$-additive cyclic, additive constacyclic and additive dual codes and find...

Let \({\mathcal {A}}\) be a \(*\)-algebra over the complex field \({\mathbb {C}}.\) For any \(T_1, T_2, \ldots , T_n \in {\mathcal {A}}\), define \(q_1(T_1)=T_1, q_2(T_1, T_2)=T_1\diamond T_2=T_1T_2^*+T_2T_1^*\) and \(q_n(T_1, T_2,\ldots , T_n)=q_{n-1}(T_1, T_2,\ldots , T_{n-1})\diamond T_n\) for all integers \(n\ge 2.\) In this article, it is show...

Assume that 𝒰 is a unital algebra over a commutative unital ring ℛ and 𝒮 is an 𝒰-bimodule. A trivial extension algebra 𝒰 ⋉ 𝒮 is defined as an ℛ-algebra with usual operations of ℛ-module 𝒰×𝒮 and the multiplication defined by (u1,s1)(u2,s2) = (u1u2,u1s2 + s1u2) for all u1,u2 ∈𝒰, s1,s2 ∈𝒮. In this paper, we prove that under certain conditions every Jo...

Let R be a commutative ring with 1 ̸ = 0, Z(R) be the set of zero-divisors of R, and Reg(R) be the set of regular elements of R. In this paper, we introduce and investigate the dot total graph of R and denote by T Z(R) (Γ(R)). It is the (undirected) simple graph with all elements of R as vertices, and any two distinct vertices x, y ∈ R are adjacent...

Suppose \(\mathbb {F}_{q}\) is a finite field with q elements and \(q=p^{t},\) where p is a prime and \(t\ge 1.\) Let \(\mathfrak {R}_{q}={\mathbb {F}}_q+u_{1}{\mathbb {F}}_q+u_{2}\mathbb F_q+u_{1}u_{2}{\mathbb {F}}_q\), where \(u_{1}^2=0\), \(u_{2}^2=0,\) \(u_{1}u_{2}=u_{2}u_{1}\) be a non-chain ring. A necessary and sufficient condition for a giv...

In this article, we give the structure of generalized Lie triple derivations of trivial extension algebras and prove that under certain appropriate assumptions every generalized Lie triple derivation of a trivial extension algebra is the sum of a generalized derivation and a linear mapping from the trivial extension algebra into its center that van...

Let T be a unital algebra with nontrivial idempotents. For any s = 0, then φ(s + t) − φ(s) − φ(t) ∈ Z(T) for all s, t ∈ T, and under some mild assumptions φ is of the form δ + τ, where δ : T → T is an additive derivation and τ : T → Z(T) is a map such that τ(p n (s 1 , s 2 ,. .. , s n)) = 0 for all s 1 , s 2 ,. .. , s n ∈ T with s 1 s 2 · · · s n =...

Let A be a standard operator algebra on a complex Banach space X, dimX > 1, and pn(T1, T2, . . . , Tn) the (n − 1)th-commutator of elements T1, T2, . . . , Tn ∈ A. Then every map ξ : A → A (not necessarily linear) satisfying ξ(pn(T1, T2, . . . , Tn)) = Σni=1 pn(T1, T2, . . . , Ti−1, ξ(Ti), Ti+1, . . . , Tn) for all T1, T2, . . . , Tn ∈ A is of th...

Let [Formula: see text] be a commutative ring and [Formula: see text] be its zero-divisors set. The zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text...

Let R be a commutative ring and Z(R) be its zero-divisors set. The zero-divisor graph of R, denoted by Γ(R), is an undirected graph with vertex set Z(R)^* = Z(R) \ {0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this article, for n = p^Rq^S where p and q are primes (p < q) and R and S are positive integers, we calculate...

In this paper, we characterize Lie (Jordan) σ -centralizers of triangular algebras. More precisely, we prove that, under certain conditions, every Lie σ -centralizer of a triangular algebra can be represented as the sum of a σ -centralizer and a central-valued mapping. Further, it is shown that every Jordan σ -centralizer of a triangular algebra is...

In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication yb=β(b)y+γ(b), where β is an automorphism and γ is a β-derivation on S so that S[y;β,γ] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;β,γ] and obtain the most stri...

Let $\mathfrak{T}$ be a factor von Neumann algebra acting on complex Hilbert space with dim($\mathfrak{T})\geq 2.$ For any $T, T_1, T_2, \dots, T_n \in\mathfrak{T},$ define $q_1(T)=T,$ $q_2(T_1, T_2)=T_1\diamond T_2=T_1T_2^\ast+T_2T_1^\ast$ and $q_n(T_1, \cdots, T_n)=q_{n-1}(T_1, \cdots, T_{n-1})\diamond T_n$ for all integers $n\geq 2.$ In this art...

In the present paper, we characterize Lie (Jordan) σ-centralizers of generalized matrix algebras. More precisely, we obtain some conditions under which every Lie σ-centralizer of a generalized matrix algebra can be expressed as the sum of a σ-centralizer and a center-valued mapping. Further, it is shown that under certain appropriate assumptions ev...

Let V be a finite-dimensional vector space. In this survey, we present the results concerning the fundamental properties of the graphs associated with finite-dimensional vector spaces.

Let 𝔄 {\mathfrak{A}} be a triangular ring and let p n ⁢ ( U 1 , U 2 , … , U n ) {p_{n}(U_{1},U_{2},\dots,U_{n})} denote the ( n - 1 ) {(n-1)} th commutator of elements U 1 , U 2 , … , U n ∈ 𝔄 {U_{1},U_{2},\dots,U_{n}\in\mathfrak{A}} . Suppose that ℕ {\mathbb{N}} is the set of nonnegative integers and 𝔏 = { ξ r } r ∈ ℕ {\mathfrak{L}=\{{\xi_{r}}\}_{r...

Let A be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan n-derivation ∆ : A → A is additive. More precisely, it is proved that ∆ is of the form ∆(t) = µt + δ(t), where µ ∈ Z(A) and δ : A → A is a Jordan n-derivation. The main result is then applied to some...

Let Rr = Fq + v1F Fq + ··· + vr Fq, where q is the power of prime, vi² = vi, vivj = vjvi = 0 for 1 ≤ i, j ≤ r and r ≥ 1. In this paper, the structure of λ-constacyclic codes over the ring Rr is studied and a Gray map ϕ from Rrn to Fq(r+1)n is given. The necessary and sufficient conditions for these codes to contain their Euclidean duals are determi...

Let $\mathcal{A}$ be a factor von Neumann algebra with $dim\mathcal{A}\geq 2.$ For any $X_1, X_2,\cdots, X_n\in\mathcal{A},$ define $p_1(X_1)=X_1,$ $p_2(X_1, X_2)=[X_1, X_2]_{\bullet}=X_1X_2^{\ast}-X_2X_1^{\ast}$ and $p_n(X_1, X_2,\cdots, X_n)=[p_{n-1}(X_1, X_2,\cdots, X_{n-1}), X_n]_{\bullet}$ for all integers $n\geq 2.$ In this article, we prove...

Let p be an odd prime, \(q=p^m\), \(R_1=\mathbb F_{q}+u{\mathbb {F}}_{q}+v{\mathbb {F}}_{q}+uv{\mathbb {F}}_{q}\) with \(u^2=1\), \(v^2=1\), \(uv=vu\) and \(R_2={\mathbb {F}}_{q}[u,v,w]/\langle u^2-1, v^2-1,w^2-1, uv-vu,vw-wv,wu-uw\rangle \) with \(u^2=1\), \(v^2=1\), \(w^2=1\), \(uv=vu\), \(vw=wv\), \(wu=uw\). In this paper, \({\mathbb {F}}_q R_1R...

Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Lie $n$-derivation $\Delta:\mathfrak{A}\rightarrow\mathfrak{A}$ is of the form $\Delta(u)=zu+\delta(u),$ where $z\in\mathcal{Z}(\mathfrak{A})$ and $\delta:\mathfrak{A}\rightarrow\mathfrak{A}$ is...

In the previous chapter, we have studied and characterized the maps V×V→F, having the property of being linear in each of their arguments, where V is a vector space over the field F.

Let R be a semiprime (or prime) ring and U be a nonzero ideal of R. In the present paper, we study the notions of multiplicative generalized α-skew derivations on ideals of R and prove that if R admits a multiplicative generalized α-skew derivation G associated with a nonzero additive map d and an automorphism α, then d is necessarily an α-skew der...

Let $\mathcal{R}$ be a commutative ring with identity, $\mathcal{A}$, $\mathcal{B}$ be $\mathcal{R}$-algebras, $\mathcal{M}$ be an $(\mathcal{A},\mathcal{B})$-bimodule and $\mathcal{N}$ be a $(\mathcal{B},\mathcal{A})$-bimodule. The $\mathcal{R}$-algebra $\mathcal{G}=G(\mathcal{A},\mathcal{M},\mathcal{N},\mathcal{B})$ is a generalized matrix algebr...

Let W be a fixed k-dimensional subspace of an n-dimensional vector space V such that n − k ≥ 1. In this paper, we introduce a graph structure, called the subspace based subspace inclusion graph I W n (V), where the vertex set V(I W n (V)) is the collection of all subspaces U of V such that U+W = V and U W, i.e., V(I W n (V)) = {U ⊆ V | U+W = V, U W...

Let R be a commutative ring with unity, A, B be R-algebras and M be an (A, B)-bimodule. Let T = T ri(A, M, B) be a (n − 1)-torsion free triangular algebra. In this article, we prove that every multiplicative Lie n-higher derivation on triangular algebras has the standard form. Also, the main result is applied to some classical examples of triangula...

Duality is a very important tool in mathematics. In this chapter, we explore some instances of duality. Let V be a vector space over a field \(\mathbb {F}\). Since every field is a vector space over itself, one can consider the set of all linear transformations \(Hom(V,\mathbb {F}).\)

In the previous chapters, we have considered vector space V over an arbitrary field \(\mathbb {F}\). In the present chapter, we shall restrict ourselves over the field of reals \(\mathbb {R}\) or the complex field \(\mathbb {C}\). One can see that the concept of “length” and “orthogonality” did not appear in the investigation of vector space over a...

A map between any two algebraic structures (say groups, rings, fields, modules or algebra) of same kind is said to be an isomorphism if it is one-to-one, onto and homomorphism; roughly speaking, it preserves the operations in the underlying algebraic structures.

If we consider the set V of all vectors in a plane (or in a 3-dimensional Euclidean space), it can be easily seen that the sum of two vectors is a vector again and under the binary operation of vector addition \('+'\), V forms an additive abelian group.

In this chapter, we shall study common problems in numerical linear algebra which includes LU and PLU decompositions together with their applications in solving a linear system of equations. Further, we shall briefly discuss the power method which gives an approximation to the eigenvalue of the greatest absolute value and corresponding eigenvectors...

The present chapter is aimed at providing background material in order to make the book as self-contained as possible. However, the basic information about set, relation, mapping etc. have been pre-assumed. Further, an appropriate training on the basics of matrix theory is certainly the right approach in studying linear algebra. Everybody knows tha...

This chapter is initially devoted to the study of subspaces of an affine space, by applying the theory of vector spaces, matrices and system of linear equations. By using methods involved in the theory of inner product spaces, we then stress practical computation of distances between points, lines and planes, as well as angles between lines and pla...

This chapter is devoted to the study of the properties of bilinear and quadratic forms, defined on a vector space V over a field \(\mathbb {F}\). The main goal will be the construction of appropriate methods aimed at obtaining the canonical expression of the functions, in terms of suitable bases for V. To do this, we will introduce the concept of o...

In this chapter, we provide a method for solving systems of linear ordinary differential equations by using techniques associated with the calculation of eigenvalues, eigenvectors and generalized eigenvectors of matrices. We learn in calculus how to solve differential equations and the system of differential equations. Here, we firstly show how to...

In Chapter 7, bilinear and quadratic forms with various ramifications have been discussed. In the present chapter, we address an aesthetic concern raised by bilinear forms, and as a part of this study, the tensor product of vector spaces has been introduced. Further, besides the study of tensor product of linear transformations, in the subsequent s...

In this chapter, we study the structure of linear operators. In all that follows, V will be a finite dimensional vector space and \(T:V\rightarrow V\) a linear operator from V to itself. We recall that the kernel N(T) and the image R(T) of T are both subspaces of V and, in the light of the rank-nullity theorem, the following conditions are equivale...

In this article, we carried out the study of multiplicative Lie skew derivation on triangular algebras having nontrivial idempotents and prove that it has standard form. As a consequence of our main result, we mention its application to nest algebras and block upper triangular matrix algebras

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 this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally...

Let R be a commutative ring with unity and let 𝒰 be a unital algebra over ℛ (or a field 𝔽). An R-linear map L : 𝒰 → 𝒰 is called a Lie derivation on 𝒰 if L([u, v]) = [L(u), v] + [u,L(v)] holds for all u, v 𝜖 𝒰. For scalar 𝜉 𝜖 𝔽, an additive map L : 𝒰 → 𝒰 is called an additive 𝜉-Lie derivation on 𝒰 if L([u, v]𝜉) = [L(u), v]𝜉 + [u,L(v)]𝜉, where [u, v]...

Let A be a von Neumann algebra with no central summands of type I_1. In this article, we study Lie n-derivation on von Neumann algebra and prove that every additive Lie n-derivation on a von Neumann algebra has standard form at zero product as well as at projection product.

Motivated by the works of Wang [Y. Wang, \textit{Lie (Jordan) derivations of arbitrary triangular algebras,} Aequationes Mathematicae, \textbf{93} (2019), 1221-1229] and Moafian et al. [F. Moafian and H. R. Ebrahimi Vishki, \textit{Lie higher derivations on triangular algebras revisited,} Filomat, \textbf{30}(12) (2016), 3187-3194.], we study Lie h...

Suppose that V is a n-dimensional vector space and W is its fixed k-dimensional subspace such that n-k≥1. In the present article, we initiate the study of a new graph structure “subspace-based subspace sum graph” GW(V), where the vertex set V(GW(V)) is the collection of all subspaces W of V such that W+W≠V and W⊈W, i.e., V(GW(V))={W⊆V|W+W≠V,W⊈W} an...

Let \({\mathcal {U}}\) be a unital algebra over a unital commutative ring \({\mathcal {R}}\) and \({\mathcal {M}}\) be a \({\mathcal {U}}\)-bimodule. A trivial extension algebra \({\mathcal {U}} < imes {\mathcal {M}}\) is defined as an \({\mathcal {R}}\)-algebra with the usual operations of the \({\mathcal {R}}\)-linear space \({\mathcal {U}}\oplus...

Let N be the set of nonnegative integers and A be a (n-1)-torsion free triangular algebra over a commutative ring R. In the present paper, under some mild assumptions on A, it is prove that if δ:A→A is an R-linear mapping satisfying δ(pn(X1,X2,⋯,Xn))=∑i=1i=npn(X1,X2,⋯,Xi-1,δ(Xi),Xi+1,⋯,Xn) for all X1,X2,⋯,Xn∈A with X1X2=0 (resp. X1X2=P, where P is...

In this paper, we give a description of Lie (Jordan) triple derivations and generalized Lie (Jordan) triple derivations of an arbitrary triangular algebra \({\mathfrak {A}}\) through a triangular algebra \({\mathfrak {A}}^{0},\) where \({\mathfrak {A}}^{0}\) is a triangular algebra constructed from the given triangular algebra \({\mathfrak {A}}\) u...

In this paper, we study cyclic codes over a finite non-chain ring \(\mathbb {F}_{p^{m}}+v\mathbb {F}_{p^{m}}\) with v2 = 1 and find new and better quantum error-correcting codes than previously known quantum error correcting codes over \(\mathbb {F}_{p^{m}}\). Therewith, we characterize the LCD codes and obtain many new LCD codes. Also, we prove th...

Let A be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I. Let pn(X1,X2,⋯,Xn) be the polynomial defined by n indeterminates X1,⋯,Xn and their multiple *-Lie products and N be the set of non-negative integers. In this paper, it is shown that if A is closed under the adjoint operation and D...

Let \({\mathcal {R}}\) be a ring with the center \({\mathcal {Z}}({\mathcal {R}})\) containing a nontrivial idempotent. Suppose \(p_n(X_1,X_2,\dots , X_n)\) is the polynomial defined by n noncommuting indeterminates \(X_1, \dots , X_n\) and their multiple Lie products. In this article, under a lenient condition on \({\mathcal {R}}\), it is shown th...

Let R be a commutative ring with unity. The R-algebra G is a generalized matrix algebra defined by the Morita context. In this article, we study multiplicative generalized Lie type derivations on generalized matrix algebras and prove that it has the standard form.

Let SG= G(A,M,N, B) be a generalized matrix algebra over a commutative ring with unity. In the present article, we study k-semi-centralizing maps of generalized matrix algebras

Let R be a commutative ring with unity and U be an unital algebra over R (or field F). An R-linear map L:U ↦U is called a Lie derivation on U if L([u,v])=[L(u),v]+[u,L(v)] holds for all u,v ∊U. For scalar ξ ∊F an additive map L:U ↦U is called an additive ξ-Lie derivation on U if L([u,v]ξ)=[L(u),v]ξ+[u,L(v)]ξ, where [u,v]ξ = uv-ξvu holds for all u,v...

Let N be the set of nonnegative integers and A be a 2-torsion free triangular algebra over a commutative ring R. In the present paper, under some lenient assumptions on A, it is proved that if ∆ ={δn}n∈N is a sequence of R-linear mappings δn:A→A satisfying δn([[x,y],z]) =∑i+j+k=n[[δi(x),δj(y)],δk(z)] for all x,y,z∈A with xy= 0 (resp.xy=p, where p i...

Let [Formula: see text] be a ∗-ring with the center 𝒵([Formula: see text]) and ℕ be the set of nonnegative integers. In this paper, it is shown that if [Formula: see text] contains a nontrivial self-adjoint idempotent which admits a generalized ∗-Lie higher derivable mapping Δ = {G n } n∈ℕ associated with a ∗-Lie higher derivable mapping ℒ = {L n }...

Let R be a semiprime ring with an involution ‘’. Let and denote its right Utumi quotient ring and right symmetric Martindale quotient ring, respectively. In the present paper, the following extension problems have been obtained: (i) an involution of a semiprime ring can be uniquely extended to its right symmetric Martindale quotient ring; (ii) if R...

We extend the domain of applicability of the concept of (1, 𝛼)-derivations in 3-prime near-rings by analyzing the structure and commutativity of the near-rings admitting (1, 𝛼)-derivations satisfying certain differential identities.

Let {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}} . In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m -pot...

In this paper, we study the structural properties of (α+u1β+u2γ+u1u2δ)-constacyclic codes over R=Fq[u1,u2]/〈u12−u1,u22−u2,u1u2−u2u1〉 where q=pm for odd prime p and m≥1. We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4n. Als...

In this paper, we obtain that under certain assumptions every nonlinear ∗-Lie derivation L:U→U is of standard form, i.e.; L has the form L=d+τ , where d:U→U is an additive ∗-derivation and τ:U→Z(U) is central mapping vanishing at commutators.

In this article, we study generalized derivation and generalized Jordan derivation on generalized matrix algebras and prove that every generalized Jordan derivation can be written as the sum of a generalized derivation and antiderivation with some limitations. Also, we show that every generalized Jordan derivation is a generalized derivation on tri...

Let be a ring containing a nontrivial idempotent with the center Z() and ℕ be the set of all non-negative integers. Let Δ = {Gn}n∈ℕ be a family of mappings Gn : (not necessarily additive) such that , the identity mapping of . Then Δ is said to be a generalized Lie triple higher derivable mapping of holds for all a; b; c ∈ and for each n ∈ ℕ, where...

Let R be a commutative ring with unity and G=G(A,M,N,B) be a generalized matrix algebra. In this article, we give the structure of Lie triple derivation L on a generalized matrix algebra G and prove that under certain appropriate assumptions L on G is proper, i.e., L=δ+χ, where δ is a derivation on G and χ is a mapping from G into its center Z(G) w...

This book is a collection of selected research papers, some of which were presented at the International Conference on Differential Geometry, Algebra and Analysis (ICDGAA 2016), held at the Department of Mathematics, Jamia Millia Islamia, New Delhi, from 15–17 November 2016. It covers a wide range of topics—geometry of submanifolds, geometry of sta...

Let R be a commutative ring with unity, A, B be R-algebras, M be (A, B)-bimodule and N be (B, A)-bimodule. The R-algebra G=G(A, M, N, B) is a generalized matrix algebra defined by the Morita context (A, B, M, N, ξMN, ΩNM). In this article, we study Jordan σ-derivations on generalized matrix algebras.

Let R be a ring and N be the set of all non-negative integers. Let D = {dn}n∈N be a family of mappings dn : R → R such that d0 = IR, the identity map of R satisfying dn(ab + ba) = i+j=n di(a)dj(b) + di(b)dj(a) for all a, b ∈ R and for each n ∈ N. In the present paper it is shown that if a family ∆ = {Gn}n∈N of mappings Gn : R → R satisfies Gn(ab+ba...

Let R be a commutative ring with unity, A = Tri(A, M, B) be a triangular algebra consisting of unital algebras A, B and (A, B)-bimodule M which is faithful as a left A-module and also as a right B-module. Let σ and τ be two automorphisms of A. A family ∆ = {δ n } n∈N of R-linear mappings δ n : A → A is said to be a generalized Jordan triple (σ, τ)-...

Let p be an odd prime, and k be an integer such that gcd(k,p) = 1. Using pairwise orthogonal idempotents γ1,γ2,γ3 of the ring ℛ = 𝔽p[u]/〈uk+1 − u〉, with γ1 + γ2 + γ3 = 1, ℛ is decomposed as ℛ = γ1ℛ⊕ γ2ℛ⊕ γ3ℛ, which contains the ring R = γ1𝔽p ⊕ γ2𝔽p ⊕ γ3𝔽p as a subring. It is shown that, for λ0,λk ∈ 𝔽p, λ0 + ukλ k ∈ R, and it is invertible if and on...

Let {\mathcal{R}} be a prime ring with center {Z(\mathcal{R})} and {*} an involution of {\mathcal{R}} . Suppose that {\mathcal{R}} admits generalized derivations F , G and H associated with a nonzero derivation f , g and h of {\mathcal{R}} , respectively. In the present paper, we investigate the commutativity of a prime ring {\mathcal{R}} satisfyin...

Let p be an odd prime, and k be an integer such that gcd(k, p) = 1. Using pairwise orthogonal idempotents γ 1 , γ 2 , γ 3 of the ring R = F p [u]/u k+1 − u, with γ 1 + γ 2 + γ 3 = 1, R is decomposed as R = γ 1 R ⊕ γ 2 R ⊕ γ 3 R, which contains the ring R = γ 1 F p ⊕ γ 2 F p ⊕ γ 3 F p as a subring. It is shown that, for λ 0 , λ k ∈ F p , λ 0 + u k λ...

Let be a von Neumann algebra without nonzero central abelian projections on a complex Hilbert space . Let pn(X1, X2, · · ·, Xn) be the polynomial defined by n indeterminates X1, · · ·, Xn and their Jordan multiple ∗-products. In this paper it is shown that a family 𝒟 = {dm}m∈ℕ of mappings such that , the identity map on satisfies the condition for...

In this paper, we study a class of skew constacyclic codes over the ring \(R=F_q+u_1F_{q}+\cdots +u_{2m}F_{q}\), where \(u_i^2=u_i\), \(u_iu_j=u_ju_i=0\), for \(i,j=1,2,\ldots ,2m ~,~ i \ne j\) and \(q=p^s\), and derive the generator polynomials of this class of codes over R. Also, by using Calderbank–Shor–Steane construction, some new non-binary q...

Let R be a commutative ring with unity. The R-algebra A = T ri(A 11 , A 12 , A 22) is a triangular algebra consisting of algebras A 11 and A 22 over R and A 12 a (A 11 , A 22)-bimodule, which is faithful as a left A 11-module and also as a right A 22-module. In this paper, it is shown that every r-multiplicative bijective map, r-Jordan bijective ma...

In this paper, quantum codes over Fp from cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉, where u2 = 1, v3 = v, uv = vu and p is an odd prime have been studied. We give the structure of cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉 and obtain quantum codes over Fp using self-orthogonal property of these classes of cod...

Let R be a ∗-ring with the center Z(R). In the present paper we study commutativity of a prime ring which admits derivations d, g satisfying any one of the properties: (i) d([x, x∗]) ± [x, x∗] ∈ Z(R), (ii) d(x ◦ x∗) ± x ◦ x∗ ∈ Z(R), (iii) d([x, x∗]) ± x ◦ x∗ ∈ Z(R), (iv) d(x ◦ x∗) ± [x, x∗] ∈ Z(R), (v) d(x) ◦ d(x∗) ± x ◦ x∗ ∈ Z(R), (vi) d(x)g(x∗) ±...

In the present paper, we investigate the commutativity of 3-prime near-rings satisfying certain conditions involving left generalized multiplicative derivations on semigroup ideals. Moreover, examples have been provided to justify the necessity of 3-primeness condition in the hypotheses of various results.

Let \(\mathfrak {R}\) be a prime ring with characteristic different from 2 and m, n, k be fixed positive integers. In this paper we study the case when \(\mathfrak {R}\) admits a generalized derivation \(\mathscr {F}\) with associated derivation \(\mathfrak {D}\) such that \((i)~(\mathscr {F}(x)\circ \mathscr {F}(y))^k=\mathscr {F}(x\circ _ky)~(ii)...

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.