On Automorphisms and Commutativity in Semiprime Rings

2013 ◽  
Vol 56 (3) ◽  
pp. 584-592 ◽  
Author(s):  
Pao-Kuei Liau ◽  
Cheng-Kai Liu
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Positive Integers

Abstract. Let R be a semiprime ring with center Z(R). For x, y ∊ R, we denote by [x, y] = xy – yx the commutator of x and y. If σ is a non-identity automorphism of R such thatfor all x ∊ R, where n0, n1, n2, … nk are fixed positive integers, then there exists a map μ: R → Z(R) such that σ(x) = x + μ(x) for all x ∊ R. In particular, when R is a prime ring, R is commutative.

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.

AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350092 ◽  
Author(s):  
CHENG-KAI LIU
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Engel Condition ◽  
Positive Integers

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

scholarly journals Generalized derivations in prime and semiprime

2015 ◽  
Vol 34 (2) ◽  
pp. 29
Author(s):  
Shuliang Huang ◽  
Nadeem Ur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Positive Integers

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$  fixed positive integers.  If $R$ admits a generalized derivation $F$ associated with a  nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for  all $x,y\in I$, then $R$ is commutative. Moreover  we also examine the case when $R$ is a semiprime ring.

On commuting additive mappings on semiprime rings

2020 ◽  
pp. 2150079
Author(s):  
Siriporn Lapuangkham ◽  
Utsanee Leerawat
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Semiprime Rings ◽  
Additive Mappings

The main purpose of this paper is to describe the structure of a pair of additive mappings that are commuting on a semiprime ring. Furthermore, we prove that the existence of different commuting epimorphisms on a prime ring forces the ring to be commutative. Finally, we characterize additive mappings, which act as homomorphisms or antihomomorphisms on a semiprime ring.

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.

Semiprime Rings with Hypercentral Derivations

1995 ◽  
Vol 38 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Left Ideal ◽  
Semiprime Ring ◽  
Quotient Ring ◽  
Central Idempotent ◽  
Utumi Quotient Ring ◽  
Semiprime Rings ◽  
Positive Integers

AbstractLetRbe a semiprime ring with a derivationd, λ a left ideal ofRandk, ntwo positive integers. Suppose that[d(xn),xn]k= 0 for allx∊ λ. Then [λ,R]d(R)= 0. That is, there exists a central idempotente∊U, the left Utumi quotient ring ofR, such thatdvanishes identically oneUand λ(l —e) is central in (1 —e)U

On Prime and Semiprime Rings with Derivations

2006 ◽  
Vol 13 (03) ◽  
pp. 371-380 ◽  
Author(s):  
Nurcan Argaç
Keyword(s):  
Prime Ring ◽  
Nonempty Subset ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

A note on an Engel condition with derivations in rings

2017 ◽  
Vol 26 (1) ◽  
pp. 19-27
Author(s):  
Mohd Arif Raza ◽  
◽  
Nadeem Rehman ur ◽  
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Extended Centroid ◽  
Engel Condition ◽  
Positive Integers

Let $R$ be a prime ring with center $Z(R)$, $C$ the extended centroid of $R$, $d$ a derivation of $R$ and $n,k$ be two fixed positive integers. In the present paper we investigate the behavior of a prime ring $R$ satisfying any one of the properties (i)~$d([x,y]_k)^n=[x,y]_k$ (ii) if $char(R)\neq 2$, $d([x,y]_k)-[x,y]_k\in Z(R)$ for all $x,y$ in some appropriate subset of $R$. Moreover, we also examine the case when $R$ is a semiprime ring

scholarly journals Remarks on Generalized Derivations in Prime and Semiprime Rings

2010 ◽  
Vol 2010 ◽  
pp. 1-6 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

LetRbe a ring with centerZandIa nonzero ideal ofR. An additive mappingF:R→Ris called a generalized derivation ofRif there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)for allx,y∈R. In the present paper, we prove that ifF([x,y])=±[x,y]for allx,y∈IorF(x∘y)=±(x∘y)for allx,y∈I, then the semiprime ringRmust contains a nonzero central ideal, providedd(I)≠0. In caseRis prime ring,Rmust be commutative, providedd≠0. The cases (i)F([x,y])±[x,y]∈Zand (ii)F(x∘y)±(x∘y)∈Zfor allx,y∈Iare also studied.

scholarly journals Jordan ?-Centralizers of Prime and Semiprime Rings

2010 ◽  
Vol 7 (4) ◽  
pp. 1426-1431
Author(s):  
Baghdad Science Journal
Keyword(s):  
Prime Ring ◽  
Zero Divisor ◽  
Semiprime Ring ◽  
Additive Mapping ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

The purpose of this paper is to prove the following result: Let R be a 2-torsion free ring and T: R?R an additive mapping such that T is left (right) Jordan ?-centralizers on R. Then T is a left (right) ?-centralizer of R, if one of the following conditions hold (i) R is a semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring, where ? be surjective endomorphism of R . It is also proved that if T(x?y)=T(x)??(y)=?(x)?T(y) for all x, y ? R and ?-centralizers of R coincide under same condition and ?(Z(R)) = Z(R) .

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