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We study the primeness of noncommutative polynomials on prime rings. Let R be a prime ring with extended centroid C, ρ a right ideal of R, f(X1,…,Xt) a noncommutative polynomial over C, which is not a polynomial identity (PI) for ρ, and a,b ∈ R∖{0}. Then af(x1,…,xt)b = 0 for all x1,…,xt ∈ ρ if and only if one of the following holds: (i) aρ = 0; (ii) ρC = eRC for some idempotent e ∈ RC and b ∈ ρC such that either f(X1,…,Xt)Xt+1 is a PI for ρ or f(X1,…,Xt) is central-valued on eRCe and ab = 0. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of X-primeness of rings. [ABSTRACT FROM AUTHOR]

Let R be an associative ring. Given a positive integer n ≥ 2 , for a 1 , ... , a n ∈ R we define [ a 1 , ... , a n ] n : = a 1 a 2 ⋯ a n − a n a n − 1 ⋯ a 1 , the n -generalized commutator of a 1 , ... , a n . By an n -generalized Lie ideal of R (at the (r + 1) th position with r ≥ 0) we mean an additive subgroup A of R satisfying [ x 1 , ... , x r , a , y 1 , ... , y s ] n ∈ A for all x i , y j ∈ R and all a ∈ A , where r + s = n − 1. In the paper, we study n -generalized commutators of rings and prove that if R is a noncommutative prime ring and n ≥ 3 , then every nonzero n -generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is a noncommutative simple ring, then R = [ R , ... , R ] n . This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on n -generalized commutators and their relationship with noncommutative polynomials are also discussed. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with τ an anti-endomorphism or endomorphism. It is proved that if τ satisfies an Engel condition for all , where are k fixed positive integers, then τ is a commuting map (i.e. for all). The theorem generalizes the results proved in [22] with τ an anti-automorphism of finite order and in [10, 11] with R a division ring, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we characterize Lie ideals, which are either finitely generated ℤ -modules or maximal, in (centrally closed) prime rings. As consequences, we extend the results proved in [1] for finite dimensional central division algebras of characteristic not 2 to simple rings of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

LetRbe a simple GPI-ring (i.e., a simple ring satisfying a nontrivial generalized polynomial identity) with a nonzero derivationδand with Martindale symmetric ring of quotientsQ. Motivated by the Noether-Skolem theorem, we characterize linear differential mapsforx∈R, whereare finitely many elements inQ, such thatϕ(R)⊆[R,R]. The result is described and proved in terms of polynomials inQ[t;δ], the Ore extension ofQby the derivationδ. [ABSTRACT FROM AUTHOR]

We study bi-additive maps ofξ-Lie product type satisfying certain local property from the viewpoint of functional identities. LetRbe a unital prime ring with extended centroidC,ξ∈C, and maximal left ring of quotientsQml(R). Suppose thatRcontains a nontrivial idempotent. We completely characterize additive mapsϕ:R→Qml(R) having the property thatwheneverx,y∈Rsatisfyxy = 0 = yx. As consequences, some known results are fully generalized. [ABSTRACT FROM AUTHOR]

LetRbe a noncommutative prime ring with extended centroidCand maximal left ring of quotientsQml(R). The aim of the paper is to study a basic functional identity concerning bi-additive maps onR. Precisely, it is proved that a bi-additive mapB:R×R→Qml(R) satisfying [B(x,y),[x,y]] = 0 for allx,y∈Rmust be of the form (x,y)↦λ[x,y]+μ(x,y) forx,y∈R, where λ∈Candμ:R×R→Cis a bi-additive map. As applications to the theorem, Jordanσ-biderivations withσan epimorphism and additive commuting maps on noncommutative Lie ideals ofRare characterized. [ABSTRACT FROM AUTHOR]

Letbe a non-commutative prime ring withits maximal right ring of quotients. It is proved that the functional identityfor bi-additive mapsonly has the trivial solution. Basing on the result, we characterize Jordan-derivations and generalized Jordan semiderivations in terms of derivations,-derivations and semiderivations. [ABSTRACT FROM AUTHOR]

Let R be a prime ring which is not commutative, with maximal symmetric ring of quotients Q ms (R), and let τ be an anti-automorphism of R. An additive map δ: R → Q ms (R) is called a Jordan τ-derivation if δ(x 2) = δ(x)x τ + xδ(x) for all x ∈ R. A Jordan τ-derivation of R is called X-inner if it is of the form x → ax τ - xa for x ∈ R, where a ∈ Qms(R). It is proved that any Jordan τ-derivation of R is X-inner if either R is not a GPI-ring or R is a PI-ring except when charR = 2 and dim C RC = 4, where C is the extended centroid of R. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether-Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion. [ABSTRACT FROM AUTHOR]

Letbe a prime ring, which is not commutative, with involutionand with maximal symmetric ring of quotients. An additive mapis called a Jordan-derivation iffor all. A Jordan-derivation ofis called X-inner if it is of the formfor, where. Based on a recent result of Lee and Zhou, we prove that every Jordan-derivation ofis X-inner except whenand, whereis the extended centroid of. Non-X-inner Jordan-derivations are also constructed for the exceptional case. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z ( R ) and with extended centroid C . We give a complete characterization of Jordan derivations of R when char R = 2 and dim C ⁡ R C = 4 : An additive map δ : R → R C is a Jordan derivation if and only if there exist a derivation d : R → R C and an additive map μ : R → C such that δ = d + μ and μ ( x 2 ) = 0 for all x ∈ R . As consequences, it is proved among other things: Any Z ( R ) -linear Jordan derivation of R is a derivation if dim C ⁡ R C < ∞ . Moreover, if C is either a finite field or an algebraically closed field, where char C = 2 and n ≥ 2 , then every Jordan derivation of M n ( C ) is a derivation. [ABSTRACT FROM AUTHOR]

LetRbe a semiprime ring withQml(R) the maximal left ring of quotients ofR. Suppose thatT:R → Qml(R) is an additive map satisfyingT(x2) = xT(x) for allx ∈ R. ThenTis a right centralizer; that is, there existsa ∈ Qml(R) such thatT(x) = xafor allx ∈ R. [ABSTRACT FROM AUTHOR]

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a semiprime ring with extended centroid $C$ and with maximal right ring of quotients $Q_{mr}(R)$. Let $d{:}\ R\to Q_{mr}(R)$ be an additive map and $b\in Q_{mr}(R)$. An additive map $\delta {:}\ R\to Q_{mr}(R)$ is called a (left) $b$-generalized derivation with associated map $d$ if $\delta (xy)=\delta (x)y+bxd(y)$ for all $x, y\in R$. This gives a unified viewpoint of derivations, generalized derivations and generalized $\sigma $-derivations with an X-inner automorphism $\sigma $. We give a complete characterization of $b$-generalized derivations of $R$ having nilpotent values of bounded index. This extends several known results in the literature. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a prime ring, C its extended centroid and (resp. Q) its left (resp. symmetric) Martindale quotient ring. Let δ be a σ-derivation of R, where σ is an automorphism of R. We show the equivalence of K-polynomials (resp. K-identities) of δ and cv-polynomials (resp. semi-invariant polynomials) in the Ore extension . We prove the existence of K-polynomials of δ in certain rather general family of maps. As applications, the following are proved among other things: Consider the expression , where and . [(1)] If then either R is a GPI-ring or for all . [(2)] If then either R is commutative or for all . [Copyright &y& Elsevier]

Abstract: Let R be a prime ring with extended centroid C and symmetric Martindale quotient ring . Suppose that contains a nontrivial idempotent e such that . Let be the bi-additive map , where are additive maps and where are fixed. Suppose that ϕ is zero-product preserving, that is, for with . Then there exists a derivation such that both G and H are equal to δ plus elementary operators. Moreover, there is an additive map such that for all . The result is a natural generalization of several related theorems in the literature. Actually we prove some more general theorems. [Copyright &y& Elsevier]

Abstract: Let be a prime ring with extended centroid and let be a reduced differential polynomial with coefficients in , the symmetric Martindale quotient ring of , and with zero constant term. Let and . We prove that the finiteness of and the finite-dimensionality of the -span of are equivalent to that of and that of the -span of , respectively. Hence some questions on differential polynomials are reduced to those on ordinary generalized polynomials. Let and be two derivations of a Lie ideal of and a right ideal of . As applications of our theorems, we obtain the necessary and sufficiency conditions for the finiteness of and and for the finite-dimensionality of the -spans of and . [Copyright &y& Elsevier]

Let R be a prime ring with extended centroid F and δ an F-algebraic derivation of R with the associated inner derivation ad(b). Set [image omitted], the subring of constants of δ in R. We characterize the primeness and semiprimeness of R(δ) by the minimal polynomial μ(λ) of b over F. We also show that, if R(δ) is a prime PI-ring, then PI-deg(R) = PI-deg(R(δ)) × degFb, where PI-deg(R) denotes the PI-degree of R. This is also extended to the case when R(δ) is a semiprime ring. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a prime ring with extended centroid C. Let a ∈ R be C-algebraic. We compute the minimal polynomial of the inner derivation ad(a) (resp. the inner automorphism I a for unit a) in terms of the minimal polynomial of a. Chung and Luh proved that the index of nilpotency of a nilpotent derivation is an odd number if char R ≠2 and is a 2-power if char R =2. As an application, we generalize this to arbitrary algebraic derivations and automorphisms. [Copyright &y& Elsevier]

Let R be a prime ring and d a derivation of R. In the ring of additive endomorphisms of the Abelian group (R, +), let S be the subring generated by aLdm, where a ∈ R and m ≥ 0 and where aL:x ∈ R ↦ ax ∈ R for a ∈ R. We compute the prime radical and minimal prime ideals of S via the skew polynomial ring R[x; d] by the surjective ring homomorphism [image omitted] We compute explicitly the kernel A of ϕ, the prime radical P over A and minimal prime ideals over A (Theorem 2). We obtain a necessary and sufficient condition for S to be simple, prime or semiprime (Corollary 3). As an application, let d be nilpotent. We show that the d-extension of R defined in Grzeszczuk (1992) is canonically isomorphic to the quotient ring of S modulo its prime radical (Corollary 14). [ABSTRACT FROM AUTHOR]

Let R be a prime algebra of characteristic not 2, and let d be a q -skew s-derivation of R . We show that if d 2 n = 0 and q n ? -1, then d 2 n -1 = 0. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a prime ring with extended centroid C and d, a continuous skew derivation of R. We define the notion of K-polynomials which, in the case that d is an ordinary derivation, reduces to polynomials of the form , where . It is shown that all generalized identities with d are consequences of GPIs of R and an identity in the form , where is a K-polynomial of minimal possible order m. [Copyright &y& Elsevier]

Let R be a prime ring with extended centroid C. If a non-nilpotent q-skew σ-derivation is C-algebraic, we prove that its automorphism σ must also be C-algebraic. [Copyright &y& Elsevier]

Smyth proved that given an element x in a semiprime Banach algebra B, xB is finite-dimensional if and only if Bx is finite-dimensional. In a recent paper, Yood extended the result to arbitrary semiprime algebras. In this paper we study unital one-sided ideals in semiprime algebras from different viewpoints and prove several results on equalities of dimensions. Finally, we give a structure theorem of semiprime algebras whose zero subalgebras are finite-dimensional. All arguments concerning algebras can be easily modified to the case of rings. [ABSTRACT FROM AUTHOR]