Vincenzo De Filippis

• Professor (Full) at University of Messina

Let $\mathfrak{R}$ be a prime ring of characteristic different from $2$, $\mathcal{Q}_r^m$ be its maximal right ring of quotients, $\mathcal{C}$ be its extended centroid and $\omega(s_1,\ldots,s_n)$ be a noncentral multilinear polynomial over $\mathcal{C}$. Suppose that $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are three $X$-generalized...

Let [Formula: see text] be a prime ring of characteristic different from [Formula: see text], [Formula: see text] its Utumi quotient ring, [Formula: see text] its extended centroid, [Formula: see text] a noncentral multilinear polynomial over [Formula: see text], [Formula: see text] and [Formula: see text] two generalized derivations of [Formula: s...

Let R be a prime ring and L a Lie ideal of R. The purpose of this paper is to classify generalized derivations \(F_1,\) \(F_2,\) \(F_3\) of R satisfying the following differential identity where \(\bot\) represents either the Lie product [., .], or the Jordan product \(\circ\). Furthermore, as an application, the same identities are studied locally...

Let [Formula: see text] be a ring of characteristic different from [Formula: see text], [Formula: see text] fixed positive integers, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] a nonzero generalized skew derivation of [Formula: see text]. We prove the following results: (a) If [Formula: see text] is pri...

Let ℛ {\mathcal{R}} be a prime ring of characteristic not equal to 2, let 𝒰 {\mathcal{U}} be Utumi quotient ring of ℛ {\mathcal{R}} and let 𝒞 {\mathcal{C}} be the extended centroid of ℛ {\mathcal{R}} . Let Δ be a generalized derivation on ℛ {\mathcal{R}} , and let δ 1 {\delta_{1}} and δ 2 {\delta_{2}} be derivations on ℛ {\mathcal{R}} . Let p ⁢ ( v...

Let \(\mathfrak {S}\) be a prime ring with \(char({\mathfrak {S}}) \ne 2\), \({\mathcal {Q}}_r\) its right Martindale quotient ring, \({\mathcal {C}}\) its extended centroid, L a non-central Lie ideal of \({\mathfrak {S}}\), \({\mathcal {F}}\) and \({\mathcal {G}}\) two generalized skew derivations of \({\mathfrak {S}}\). If \({\mathcal {F}}({\math...

Let R be a noncommutative prime ring of characteristic different from 2, Qr be the right Martindale quotient ring of R, and C = Z (Qr) be the extended centroid of R. Suppose that π (x1, ..., xn) is a noncentral multilinear polynomial over C, S = {π(r_1, ..., r_n) | r_1, ..., r_n ∈ R}, F, G and H are three generalized skew-derivations of R associate...

Let R be a noncommutative prime ring of characteristic different from 2, with right Martindale quotient ring Qr and extended centroid C. Let m,n ≥ 1 be fixed integers and F a nonzero generalized skew derivation of R. In this paper, we investigate the set S={F(xm)xn+xnF(xm):x∈R} and prove that its left annihilator in R is identically zero. Using the...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, F and G two non-zero generalized skew derivations of R, associated with the same automorphism α and commuting with α. In this work we describe all possible forms of F and G in the following two cases: (a) there exist a,b∈Qr and a non-central Lie ideal L...

Let R be a prime ring of characteristic different from 2, Q its right Martindale quotient ring, C its extended centroid, I a right ideal of R, \(a\in Q\), G a nonzero X-generalized skew derivation of R, \(f(x_1,\ldots ,x_n)\) a multilinear polynomial over C with n non-commuting variables, and S the set of the evaluations of \(f(x_1,\ldots ,x_n)\) o...

Let R be a prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended centroid, α be an automorphism of R, d be a skew derivation of R with associated automorphism α, F and G be two nonzero X-generalized skew derivation of R with associated term (b,α,d) and (b′,α,d), respectively, S be the set of t...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, L a non-central Lie ideal of R (i.e., L,R≠0), n≥1 a fixed integer, F and G two generalized skew derivations of R, associated with the same automorphism α of R. If (F(xy)−G(x)y)n=0, for any x,y∈L, then there exists a∈Qr such that , for any x∈R, unless wh...

Let $R$ be a prime ring, $Q_r$ its right Martindale quotient ring, $L$ a non-central Lie ideal of $R$, $n\geq 1$ a fixed integer, $F$ and $G$ two generalized skew derivations of $R$ with the same associated automorphism, $p\in R$ a fixed element. If $p\bigl(F(x)F(y)-G(y)x\bigr)^n=0$, for any $x,y \in L$, then there exist $a,c\in Q_r$ such that $F(x...

Let R be a noncommutative prime ring of characteristic different from 2, Qr be the right Martindale quotient ring of R and C=Z(Qr) be the extended centroid of R. Suppose that f(x1,…,xn) is a noncentral multilinear polynomial over C and 0≠F, G are two generalized skew-derivations of R associated to the same automorphism α. If F(u2)=F(u)G(u) for all...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R, L a non-central Lie ideal of R, a∈R a non-zero element of R, m,r≥1 and s,t≥0 fixed integers. If a(F(u)r(F(u)su+uF(u)s)F(u)t)m=0 for all u∈L, then either there exists b∈Qr...

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.

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...

Let R be a prime ring of characteristic different from 2, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, F a generalized skew derivation of R, L a non-central Lie ideal of R and \(n\ge 1\) a fixed integer such that \([F(x),y]^n=[x,F(y)]^n\), for all \(x,y \in L\). Then there exists \(\lambda \in C\) such that \(F(x)...

We introduce the definitions of g-derivations and generalized g-derivations on a ring R. The main objective of the paper is to describe the structure of a prime ring R in which g-derivations and generalized g-derivations satisfy certain algebraic identities with involution ∗, anti-automorphism and automorphism. Some well-known results concerning de...

Let R be a prime ring of characteristic different from 2, \(Q_r\) be the right Martindale quotient ring of R and \(C=Z(Q_r)\) be the extended centroid of R. Suppose that \(f(x_1,\ldots ,x_n)\) is a noncentral multilinear polynomial over C and F, G are two nonzero generalized skew-derivations of R associated to the same automorphism of R. If $$\begi...

Let R be a prime ring of characteristic different from 2 with extended centroid C, \(n\ge 1\) a fixed positive integer, \(F, G:R\rightarrow R\) two non-zero generalized skew derivations of R. (I) If \(\biggl (F(x)x\biggr )^n\in C\) for all \(x\in R\), then the following hold: (a) if F is an inner generalized skew derivation, then either \(R\subsete...

Let R be a non-commutative prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended centroid and L be a non-central Lie ideal of R Let F and G be two non-zero generalized skew derivations of R, associated with the same automorphism α and commuting with α. If F(G(x)x)=0 for all x∈L, then one of th...

Let R be a noncommutative prime ring of characteristic different from 2 and 3, C the extended centroid of R, F and G two generalized derivations of R, d a nonzero derivation of R and \(f(x_1,\ldots ,x_n)\) a multilinear polynomial over C. Suppose that I is a nonzero ideal of R and \(f(I)=\{f(x_1,\ldots ,x_n)| x_1,\ldots ,x_n\in I\}\). If \(f(x_1,\l...

Let R be a non commutative prime ring of characteristic different from 2, U be the Utumi quotient ring of R with the extended centroid C, f(x1, … , xn ) a multilinear polynomial over C which is not central valued on R, f(R) the set of all evaluations of the polynomial f (x 1, … , xn ). Suppose F and G are two nonzero generalized derivations on R and...

Let R be a prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended centroid. In this paper we define b-generalized skew derivations of prime rings. Then we describe all possible forms of two b-generalized skew derivations F and G satisfying the condition F(x)x − xG(x) = 0, for all x ∈ S, where S...

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, L a non-central Lie ideal of R, F and G two nonzero generalized derivations of R. If for all , then one of the following holds: There exist such that and , , for all ;. There exist such that and , , for all ; .

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R and f ( x 1 , … , x n ) a multilinear polynomial over C. Suppose that f ( x 1 , … , x n ) is not central valued on R, and R does not embed in M 2 ( K ) , the algebra of 2 × 2 matrices...

Let R be a prime ring of characteristic different from 2 and 3, Q r be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x – xG(x)) ⁿ = 0 for all x ∈ L, th...

In this paper we introduce the concept of weakly left cancellative semirings. Moreover we establish a connection between commutativity for this class of semirings and derivations.

Let R be a noncommutative prime ring, I a nonzero left ideal of R, L a non-central Lie ideal of R, U the left Utumi quotient ring of R and C=Z(U) the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and m,n,r,k fixed positive integers. In the present paper, we describe the structure of R and all possible for...

In this paper we introduce the concept of weakly left cancellative semir-ings. Moreover we establish a connection between commutativity for this class of semirings and derivations. MSC2010: 16Y60, 16W25, 16U80.

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 $n\geq 1$ be a fixed integer, $R$ a prime ring with its right Martindale quotient ring $Q$, $C$ the extended centroid, and $L$ a Lie ideal of $R$. Suppose that $\alpha$ is an automorphism of $R$ and $F\neq0$ a generalized skew derivation of $R$ with the associated automorphism $\alpha$. In the paper, we investigate the situation $(F(x)F(y)-yx)^...

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let φ(x, y) = [x, y] t -[x, y] m [α([x, y]),[x, y]]n [x, y] s . Thus, (a) if φ(u, v) = 0 for any u, v L, then L ⊂ Z(R); (b) if φ(u, v) Z(R) for any...

Let R be a noncommutative prime ring of characteristic different from 2, Qr the right Martindale quotient ring of R, C=Z(Qr) the extended centroid of R and f(x1,…,xn) a noncentral multilinear polynomial over C. In this paper we describe all possible forms of two generalized skew derivations F and G of R satisfying the condition F(u)2=G(u2), for all...

Let R be a ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R and F a generalized derivation with associated non-zero derivation d of R, m≥1,n≥1 and s≥1 fixed integers. Let f(x1,…,xt) be a non-zero multilinear polynomial over C in t non-commuting variables, S⊆R be any subset of R and f(S)={f(r1,…,rt):ri∈...

Let R be a prime ring of characteristic different from 2, C its extended centroid and L a non-central Lie ideal of R. If F:R→R and G:R→R are two non-zero generalized derivations of R such that [F(u),G(u)]=0 for all u∈L, then G=λF for some λ∈C except possibly when R satisfies the standard identity s4 of degree 4.

Let R be a prime ring of characteristic different from 2 with its Utumi quotient ring U and extended centroid C, f(x1,…,xn) be multilinear polynomial over C, which is not central valued on R. If d is a non-zero derivation of R and F is a non-zero generalized derivation of R such that d([F(f(r1,…,rn))f(r1,…,rn),F(f(s1,…,sn))f(s1,…,sn)])=0 for all r1...

Motivated by Kosan and Lee’s elegant paper and Liu’s systematic works, this article is targeted towards the Engel conditions with X-generalize skew derivations on Lie ideals. Let R be a prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended centroid, F an X-generalized skew derivation of R, L a...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R and L is a non-central Lie ideal of R. If [[F(u), u], F(u)] = 0 for all u ∈ L, then either there exists λ ∈ C such that F(x) = λx, for all x ∈ R or R satisfies s4(x1, . . ....

Let $R$ be a prime ring of characteristic different from $2$, $Q$ be its maximal right ring of quotients and $C$ be its extended centroid. Suppose that $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C$, $0\neq p\in R$, $F$ and $G$ be two $b$-generalized derivations of $R$. In this paper we describe all possible forms of $F$ and $G...

Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Z(R)}\) its center, \(\mathcal {C}\) its extended centroid, \(\mathcal {L}\) a Lie ideal of \(\mathcal {R}, \mathcal {F}\) a generalized skew derivation associated with a skew derivation d and automorphism \(\alpha \). Assume that there exist \(t\ge 1\) and \(m,n\ge 0\) fixed integers such that \( v...

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, F≠ 0 an b-generalized skew derivation of R, L a non-central Lie ideal of R, 0 ≠ a∈ R and n≥ 1 a fixed integer. In this paper, we prove the following two results:1.If R has characteristic different from 2 and 3 and a[ F(x) , x] ⁿ= 0...

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and L a not central Lie ideal of R. Suppose that F, G and H are generalized derivations of R, with F≠0, such that F(G(x)x−xH(x)) = 0, for any x∈L. In this paper we describe all possible forms of F, G and H.

Let R be a ring, Qr the right Martindale quotient ring of R, C the extended centroid of R, L a noncentral Lie ideal of R, F a nonzero generalized skew derivation of R, and m,n,k≥1 be fixed integers. We prove the following results:Assume R is a prime. If either char(R)=0 or char(R)>m+1 and [F(um),un]k=0 for all u∈L, then there exists λ∈C such that F...

Let R be a prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended centroid, G be a nonzero X-generalized skew derivation of R, and S be the set of the evaluations of a multilinear polynomial f(x1,…,xn) over C with n non-commuting variables. Let u,v∈R be such that uG(x)x+G(x)xv=0 for all x∈S. Th...

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, L a non-central Lie ideal of R, F and G two non-zero generalized derivations of R. If [F(u), u]G(u) = 0 for all u ∈ L, then one of the following holds: (a) there exists λ ∈ C such that F(x) = λx, for all x ∈ R; (b) R ⊆ M2 (F)...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, C its extended centroid, a,b∈R, f(x1,…,xn) a non-central multilinear polynomial over C with n non-commuting variables and G a non-zero generalized skew derivation of R. Assume a≠0, b∉C, S={f(r1,…,rn):r1,…,rn∈R} and a[b,G(x)x]=0, for all x∈S. Then one of...

Let R be a ring, α and β two automorphisms of R. An additive mapping d: R → R is called an (α, β) -derivation if d(xy) = d(x) α (y) + β (x) d(y) for any x,y ∈ R. An additive mapping G: R → R is called a generalized (α, β) -derivation if G(xy) = G(x) α (y) + β (x) d(y) for any x,y ∈ R, where d is an (α, β) -derivation of R. In this paper we introduc...

Let R be a prime ring of characteristic different from 2 with right Martindale quotient ring Q and extended centroid C. Let further be a fixed integer, a multilinear polynomial over C which is not central-valued on R. If is a nonzero generalized skew derivation of R such that for all , then either there exists such that for all , or one of the foll...

Let R be a prime ring of characteristic different from 2, with right Martindale quotient ring Qr and extended centroid C, and let f(x1,…,xn) be a multilinear polynomial over C, not central valued on R. Suppose that F and G are skew derivations of R, associated with the automorphism α, such that F(f(r1,…,rn))G(f(r1,…,rn))+G(f(r1,…,rn))F(f(r1,…,rn))...

Let ℛ be a prime ring of characteristic different from 2, 𝒬r be its right Martindale quotient ring, 𝒬 be its two-sided Martindale quotient ring and 𝒞 be its extended centroid. Suppose that ℱ, 𝒢 are additive mappings from ℛ into itself and that f(x1,…,xn) is a non-central multilinear polynomial over 𝒞 with n non-commuting variables. We prove the fol...

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x1,..., xn) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r1,..., rn): ri ∈ R} be the set...

Let $R$ be a prime ring of characteristic different from $2$, $C$ its extended centroid, $d$ a nonzero derivation of $R$, $f(x_1,\ldots,x_n)$ a multilinear polynomial over $C$, $\varrho$ a nonzero right ideal of $R$ and $m> 1$ a fixed integer such that \begin{equation*} \bigl([d(f(r_1,\ldots,r_n)),f(r_1,\ldots,r_n)]\bigr)^m=[d(f(r_1,\ldots,r_n)),f(...

Let R be a prime ring of characteristic different from 2, C its extended centroid, L a noncentral Lie ideal of R and \(m,n,t\ge 1\) fixed integers. Suppose that F is a nonzero generalized skew derivation of R such that \(u^mF(u)^tu^n=0\), for all \(u \in L\). Then \(dim_CRC=4\).

Let $R$ be a semiprime ring of characteristic different from $2$, $C$ its extended centroid, $Z(R)$ its center, $F$ and $G$ non-zero skew derivations of $R$ with associated automorphism $\alpha$ and $m,n$ are positive integers such that \begin{equation*} [F(x),G(y)]_m=[x,y]^n ~\text{for all}~x,y \in R. \end{equation*} Then $R$ is commutative.

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and let f(x1,…, xn) be a multilinear polynomial over C, not central valued on R. Suppose that F and G are non-zero generalized derivations of R and 0 ≠ u0 is an element of R such that u0F(G(f(r1,…,rn))f(r1,…,rn))=0 for all r1,…, rn ∈ R...

Let R be a prime ring of characteristic different from 2, Q r be its right Martindale quotient ring and C be its extended centroid. Suppose that F, G are generalized skew derivations of R and \({f(x_1, \ldots, x_n)}\) is a non-central multilinear polynomial over C with n non-commuting variables. If F and G satisfy the following condition: $$F(f(r_1...

Let R be a ring. A biadditive symmetric mapping D : R x R -> R is called a symmetric skew biderivation if for every x is an element of R, the map y D(x, y) is a skew derivation of R (as well as for every y is an element of R, the map x D(x, y) is a skew derivation of R) Let D : R x R be a symmetric biderivation. A biadditive symmetric mapping Delta...

Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n(u, w) ≥1 such that (F(uw) − bwu)n = 0, then one of the following statements holds: F = 0 and b = 0;R ⊆ M2(K), the ring of 2 × 2...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, and C its extended centroid. Suppose that F, G are generalized skew derivations of R, with the same associated automorphism, and f(x1,…, xn) a noncentral multilinear polynomial over C with n noncommuting variables, such that for all r1,…, rn ∈ R. Then w...

Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R, and F a generalized derivation with associated derivation d of R, m ≥ 1, n ≥ 1 two fixed integers, and 0 ≠ a ∈ R. Assume that a((F([x, y]))m − [x, y]n) = 0, for all x, y ∈ I, then one of the following statements holds: R is commutative;...

Let R be a prime ring of characteristic different from 2 and 3, Qr its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all...

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R with the same associated automorphism α, and p(x1, …, xn) is a non-central polynomial over C such that (Formula presented.) for all x, y ∈ {p(r1, …, rn): r1, …, rn ∈ R}....

Let ℛ be a prime ring of characteristic different from 2, 𝒬r the right Martindale quotient ring of ℛ, 𝒞 the extended centroid of ℛ, F, G two generalized skew derivations of ℛ and k ≥ 1 be a fixed integer. If [F(r), r]kr − r[G(r), r]k = 0 for all r ∈ ℛ, then there exist a ∈ 𝒬r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ℛ.

Let R be a prime ring of characteristic different from 2, \(Q_r\) its right Martindale quotient ring and C its extended centroid. Suppose that F is a nonzero generalized skew derivation of R, with the associated automorphism \(\alpha \), and \(p(x_1,\ldots ,x_n)\) a noncentral polynomial over C, such that $$F\biggl ([x,y]\biggr )=[F(x),\alpha (y)]+...

This book discusses recent developments and the latest research in algebra and related topics. The book allows aspiring researchers to update their understanding of prime rings, generalized derivations, generalized semiderivations, regular semigroups, completely simple semigroups, module hulls, injective hulls, Baer modules, extending modules, loca...

Let $R$ be a prime ring of characteristic diòerent from $2$ , let ${{Q}_{r}}$ be its right Martindale quotient ring, and let $C$ be its extended centroid. Suppose that $F$ is a generalized skew derivation of $R,\,L$ a non-central Lie ideal of $R,\,0\,\ne \,a\,\in \,R,\,m\,\ge \,0$ and $n,\,s\,\ge \,1$ fixed integers. If $$a{{\left( {{u}^{m}}F\left(...

Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R ! R is called a generalized derivation on R if there exists a derivation d : R ! R such that F(xy) = F(x)y+xd(y) holds for all x; y 2 R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime ring...

Let R be a ring and let g be an endomorphism of R. The additive mapping d: R → R is called a Jordan semiderivation of R, associated with g, if d(x ² ) = d(x)x + g(x)d(x) = d(x)g(x) + xd(x) and d(g(x)) = g(d(x)) for all x ∊ R. The additive mapping F: R → R is called a generalized Jordan semiderivation of R, related to the Jordan semiderivation d and...

Let R be a noncommutative prime ring with its Utumi ring of quotients U, C = Z(U) the extended centroid of R, F a generalized derivation of R and I a nonzero ideal of R. Suppose that there exists 0 not equal a is an element of R such that a(F([x, y])(n) - [x, y]) = 0 for all x, y is an element of I, where n >= 2 is a fixed integer. Then one of the...

Let \(R\) be a prime ring of characteristic different from \(2\), \(U\) its right Utumi quotient ring, \(C\) its extended centroid, \(G\) a non-zero generalized derivation of \(R\), \(a\ne 0\) be an element of \(R\), \(I\) a non-zero right ideal of \(R\) such that \(s_4(I,\ldots ,I)I\ne 0\) and \(n,k\ge 1\) fixed integers. If \(a[G([r_1,r_2]^n),[r_...

Let R be a prime ring of characteristic different from 2, Q(r) be its right Martindale quotient ring and C be its extended centroid. Suppose that G is a nonzero generalized skew derivation of R, alpha is the associated automorphism of G, f(x(1),...,x(n)) is a non-central multilinear polynomial over C with n non-commuting variables and S = {f(r(1),....

Let R be a prime ring, U the right Utumi quotient ring of R, C its extended centroid, I a non-zero right ideal of R, f (x(1),...,x(n)) a non-central multilinear polynomial over C, F, G two generalized derivations of R, m >= 1 a fixed integer. Denote f (I) the set of all evaluations of the polynomial f (x(1),...,x(n)) in I. If F (u(m)) = G(u)(m), fo...

Let $R$ be a prime ring of characteristic different from $2$, $U$ its right Utumi quotient ring, $C$ its extended centroid, $F$ and $G$ be additive maps on $R$, $f(x_1,\ldots,x_n)$ a multilinear polynomial over $C$, $I$ a non-zero right ideal of $R$. In this paper we will obtain informations about the structure of $R$ and describe the form of $F$ a...

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x]n = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R.

Let R be a prime ring of characteristic different from 2, with extended centroid C , U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R , f ( x 1 , … , x n ) a noncentral multilinear polynomial over C in n noncommuting variables, and a , b ∈ R such that a [ F ( f ( r 1 , … , r n ) ) , f ( r...