Additive n-commuting maps on semiprime rings

2019 ◽  
Vol 63 (1) ◽  
pp. 193-216
Author(s):  
Cheng-Kai Liu
Keyword(s):  
Unsolved Problem ◽  
Semiprime Ring ◽  
Affirmative Answer ◽  
Extended Centroid ◽  
Additive Map ◽  
Semiprime Rings ◽  
Ring Of Quotients ◽  
Functional Identities ◽  
Fixed Positive Integer ◽  
Commuting Maps

AbstractLet R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

b-GENERALIZED DERIVATIONS OF SEMIPRIME RINGS HAVING NILPOTENT VALUES

2014 ◽  
Vol 96 (3) ◽  
pp. 326-337 ◽  
Author(s):  
M. TAMER KOŞAN ◽  
TSIU-KWEN LEE
Keyword(s):  
Semiprime Ring ◽  
Generalized Derivation ◽  
Complete Characterization ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Additive Map ◽  
Semiprime Rings ◽  
Inner Automorphism ◽  
Ring Of Quotients

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ 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.

scholarly journals A pair of generalized derivations in prime, semiprime rings and in Banach algebras

2021 ◽  
Vol 39 (4) ◽  
pp. 131-141
Author(s):  
Basudeb Dhara ◽  
Venus Rahmani ◽  
Shervin Sahebi
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Banach Algebras ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Semiprime Rings ◽  
Commutative Banach Algebras

Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy)+G(yx))n= (xy ±yx) for all x,y ∈ I, then one ofthe following holds:(1) R is commutative;(2) n = 1 and H(x) = x and G(x) = ±x for all x ∈ R.Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.

ON n-CENTRALIZING GENERALIZED DERIVATIONS IN SEMIPRIME RINGS WITH APPLICATIONS TO C*-ALGEBRAS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250111 ◽  
Author(s):  
BASUDEB DHARA ◽  
SHAKIR ALI
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
C Algebras ◽  
Semiprime Rings ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xn – xn f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

scholarly journals On the symmetric algebra of quotients of a C*-algebra

1990 ◽  
Vol 32 (3) ◽  
pp. 377-379 ◽  
Author(s):  
Pere Ara
Keyword(s):  
Galois Theory ◽  
Semiprime Ring ◽  
Equivalence Classes ◽  
Crossed Products ◽  
Prime Rings ◽  
C Algebra ◽  
Essential Ideal ◽  
Module Homomorphism ◽  
Semiprime Rings ◽  
Ring Of Quotients

Let R be a semiprime ring (possibly without 1). The symmetric ring of quotients of R is defined as the set of equivalence classes of essentially defined double centralizers (ƒ, g) on R; see [1], [8]. So, by definition, ƒ is a left R-module homomorphism from an essential ideal I of R into R, g is a right R-module homomorphism from an essential ideal J of R into R, and they satisfy the balanced condition ƒ(x)y = xg(y) for x ∈ Iand y ∈ J. This ring was used by Kharchenko in his investigations on the Galois theory of semiprime rings [4] and it is also a useful tool for the study of crossed products of prime rings [7]. We denote the symmetric ring of quotients of a semiprime ring R by Q(R).

Ad-nilpotent Elements of Semiprime Rings with Involution

2018 ◽  
Vol 61 (2) ◽  
pp. 318-327
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Semiprime Ring ◽  
Extended Centroid ◽  
Nilpotent Elements ◽  
Semiprime Rings ◽  
Rings With Involution ◽  
Torsion Free

AbstractLet R be an n!-torsion free semiprime ring with involution * and with extended centroid C, where n > 1 is a positive integer. We characterize a ∊ K, the Lie algebra of skew elements in R, satisfying (ada)n = 0 on K. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if a, b ∊ R satisfy (ada)n = adb on R, where either n is even or b = 0, then (a − λ)[(n+1)/2] = 0 for some λ ∊ C.

Generalized Derivations Having the Same Power Values with Left Multiplications

2013 ◽  
Vol 20 (03) ◽  
pp. 369-382 ◽  
Author(s):  
Xiaowei Xu ◽  
Jing Ma ◽  
Fengwen Niu
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Ring Of Quotients ◽  
Fixed Positive Integer

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal [Formula: see text] and a noncommutative Lie ideal L, respectively.

Inclusion properties of generalized inverses in semiprime rings

2020 ◽  
pp. 2150057
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Semiprime Ring ◽  
Generalized Inverses ◽  
Extended Centroid ◽  
Von Neumann ◽  
Common Generalization ◽  
Inclusion Properties ◽  
Semiprime Rings ◽  
Von Neumann Regular

Let [Formula: see text] be a semiprime ring, not necessarily with unity, with extended centroid [Formula: see text]. For [Formula: see text], let [Formula: see text] (respectively [Formula: see text], [Formula: see text]) denote the set of all outer (respectively inner, reflexive) inverses of [Formula: see text] in [Formula: see text]. In the paper, we study the inclusion properties of [Formula: see text], [Formula: see text] and [Formula: see text]. Among other results, we prove that for [Formula: see text] with [Formula: see text] von Neumann regular, [Formula: see text] (respectively [Formula: see text]) if and only if [Formula: see text] (respectively [Formula: see text]). Here, [Formula: see text] is the smallest idempotent in [Formula: see text] such that [Formula: see text]. This gives a common generalization of several known results.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

scholarly journals Multiplicative (Generalized) Reverse Derivations on Semiprime Ring

2018 ◽  
Vol 11 (3) ◽  
pp. 717-729
Author(s):  
Asma Ali ◽  
Ambreen Bano
Keyword(s):  
Semiprime Ring ◽  
Additive Map

Let R be a semiprime ring. A mapping F : R → R (not necessarily additive) is called a multiplicative (generalized) reverse derivation if there exists a map    d : R → R (not necessarily a derivation nor an additive map) such that F(xy) = F(y)x + yd(x) for all x, y є R. In this paper we investigate some identities involving multiplicative (generalized) reverse derivation and prove some theorems in which we characterize these mappings.

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