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Let R be a semiprime ring, not necessarily with unity, and a , b ∈ R. Let I (a) (respectively, Ref (a)) denote the set of inner (respectively, reflexive) inverses of a in R. It is proved that if I (a) ∩ I (b) ≠ ∅ , then I (a) ⊆ I (b) if and only if b = a w b = b w a for all w ∈ I (a). As an immediate consequence, if ∅ ≠ I (a) = I (b) , then a = b (see Theorem 7 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl.18(7) (2019) 1950128] for rings with unity). We also give a generalization of Theorem 10 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl.18(7) (2019) 1950128] by proving that if ∅ ≠ Ref (a) ⊆ Ref (b) then a = b. [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]

Let R be a semiprime ring with center Z(R) and with extended centroid C. Suppose that τ:R→R is an anti-homomorphism such that the image of τ has small centralizer. It is proved that the following are equivalent: (1) for all x∈R; (2) x+xτ∈Z(R) for all x∈R; (3) xxτ∈Z(R) for all x∈R. In this case, there exists an idempotent e∈C such that (1−e)R is a commutative ring and the semiprime ring eR is equipped with an involution , which is induced canonically by τ. These results generalize both the involution case by Chacron [5] and the anti-automorphism case by the author [7]. [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]

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]

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]

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]

Abstract: Suppose that R is a semiprime ring which is either a PI-ring or a countable ring. Then there exists a collection of prime ideals , , such that each is invariant under all derivations of R and . [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]