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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]

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]

LetRbe a semiprime ring with an anti-automorphismτ, which is of finite order. It is proved that iffor allx∈R, wherearekfixed positive integers, thenτis a commuting map. Moreover, commuting anti-automorphisms of semiprime rings are also characterized. [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]

Letbe a division algebra finite-dimensional over its centreand letbe an involution on the matrix algebra. Suppose thatis of the second kind and. We study elementary operators in the context of closed prime algebras with involution and then characterize all sesquilinear functionals onin terms of reduced traces and a specific involution on elementary operators. [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]

LetRbe a semiprime ring with centerZ(R), extended centroidC,Uthe maximal right ring of quotients ofR, andma positive integer. Letf:R → Ube an additivem-power commuting map. Suppose thatfisZ(R)-linear. It is proved that there exists an idempotente ∈ Csuch thatef(x) = λx + μ(x) for allx ∈ R, where λ ∈Cand μ:R → C. Moreover, (1 − e)U ≅ M2(E), whereEis a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map fromRtoUis commuting. [ABSTRACT FROM PUBLISHER]

LetRbe a semiprime ring with symmetric Martindale quotient ringQ,n ≥ 2 and letf(X) = Xnh(X), whereh(X) is a polynomial over the ring of integers withh(0) = ±1. Then there is a ring decompositionQ = Q1⊕Q2⊕Q3such thatQ1is a ring satisfyingS2n−2, the standard identity of degree 2n − 2,Q2 ≅ Mn(E) for some commutative regular self-injective ringEsuch that, for some fixedq > 1,xq = xfor allx ∈ E, andQ3is a both faithfulS2n−2-free and faithfulf-free ring. Applying the theorem, we characterizem-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with extended centroid C and with Q its Martindale symmetric ring of quotients. Suppose that δ: R → R is a C-integral derivation. For a subring A of Q, let A(δ)denote the subring of constants of δ in A. We prove that R(δ)and Q(δ)satisfy the same polynomial identities with coefficients in C. In particular, R(δ)is not nil of bounded index. [ABSTRACT FROM PUBLISHER]

For a polynomialwith zero constant term, a semiprime K-algebra R is called faithful f-free if every nonzero ideal of R does not satisfy f. We prove that a semiprime algebra has an essential ideal which is the direct sum of its largest faithful f-free ideal and its largest ideal satisfying the identity f. Here, faithful f-free algebras are characterized, and applications to some interesting differential identities are discussed. Especially with f = S2n(n ≥ 1), a description of semiprime rings is obtained in terms of faithful S2n-free rings and semiprime PI-rings of degree 2n. Semiprime PI-rings of degree 2n are realized through faithful S2n-rings. Finally, faithful S2n-rings are characterized. [ABSTRACT FROM PUBLISHER]

The goal of this note is to characterize strong commutativity preserving maps on prime or semiprime rings without the additivity assumption on the maps considered in the literature. [ABSTRACT FROM PUBLISHER]

By testing quotient rings, we give another viewpoint concerning the relationship between PI and Goldie properties, etc., and f-radical extensions of rings. The main result proved here is as follows: Let R be a prime algebra without nonzero nil right ideals. Suppose that R is f-radical over a subalgebra A, where f(X1,..., Xt) is a multilinear polynomial, not an identity for p × p matrices in case char R = p > 0. Suppose that f is not power-central valued in R. Then the maximal ring of right (left) quotients of A coincides with that of R. Moreover, R is right Goldie if and only if A is. [ABSTRACT FROM AUTHOR]

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]

We prove the Skolem-Noether Theorem for semiprime rings satisfying a strict identity: Let R be a centrally closed semiprime ring satisfying a strict identity. Suppose that σ is a C-linear automorphism of R. Then σ is X-inner and so is every C-linear σ-derivation of R. [ABSTRACT FROM AUTHOR]

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]

We show that an algebra must be finite-dimensional if it possesses a finite-dimensional maximal subalgebra. [ABSTRACT FROM AUTHOR]

Let R be a semiprime algebra over a field F, and let I be either an ideal of R or a right ideal with zero left annihilator in R. Let and , where ai, bi ∈ R. Suppose that dim F B1 < ∞(|B1| < ∞). Then dim F B2 < ∞ and dim F B1 = dim F B2 (resp.|B2| < ∞ and|B1| = |B2|). [ABSTRACT FROM AUTHOR]

Let R be a prime ring and U(R) the group of units of R. We prove that if U(R) generates R and satisfies a group identity, then R is either a domain or a full matrix ring over a finite field. [ABSTRACT FROM AUTHOR]

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]