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Commutativity of associative rings through a Streb's classification

Mohammad Ashraf — 1997

Archivum Mathematicum

Let m 0 , r 0 , s 0 , q 0 be fixed integers. Suppose that R is an associative ring with unity 1 in which for each x , y R there exist polynomials f ( X ) X 2 Z Z [ X ] , g ( X ) , h ( X ) X Z Z [ X ] such that { 1 - g ( y x m ) } [ x , x r y - x s f ( y x m ) x q ] { 1 - h ( y x m ) } = 0 . Then R is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of x and y . Finally, commutativity of one sided s-unital ring is also obtained when R satisfies some related ring properties.

A commutativity theorem for associative rings

Mohammad Ashraf — 1995

Archivum Mathematicum

Let m > 1 , s 1 be fixed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p ( x ) 0 , q = q ( x ) 0 , n = n ( x ) 0 , r = r ( x ) 0 such that either x p [ x n , y ] x q = x r [ x , y m ] y s or x p [ x n , y ] x q = y s [ x , y m ] x r for all y R . In the present paper it is shown that R is commutative if it satisfies the property Q ( m ) (i.e. for all x , y R , m [ x , y ] = 0 implies [ x , y ] = 0 ).

On left ( θ , ϕ ) -derivations of prime rings

Mohammad Ashraf — 2005

Archivum Mathematicum

Let R be a 2 -torsion free prime ring. Suppose that θ , φ are automorphisms of R . In the present paper it is established that if R admits a nonzero Jordan left ( θ , θ ) -derivation, then R is commutative. Further, as an application of this resul it is shown that every Jordan left ( θ , θ ) -derivation on R is a left ( θ , θ ) -derivation on R . Finally, in case of an arbitrary prime ring it is proved that if R admits a left ( θ , φ ) -derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of R , then d = 0 ...

Commutativity theorems for rings with differential identities on Jordan ideals

L. OukhtiteA. MamouniMohammad Ashraf — 2013

Commentationes Mathematicae Universitatis Carolinae

In this paper we investigate commutativity of ring R with involution ' * ' which admits a derivation satisfying certain algebraic identities on Jordan ideals of R . Some related results for prime rings are also discussed. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

( σ , τ ) -derivations on prime near rings

Mohammad AshrafAsma AliShakir Ali — 2004

Archivum Mathematicum

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation...

On ( σ , τ ) -derivations in prime rings

Mohammad AshrafNadeem-ur-Rehman — 2002

Archivum Mathematicum

Let R be a 2-torsion free prime ring and let σ , τ be automorphisms of R . For any x , y R , set [ x , y ] σ , τ = x σ ( y ) - τ ( y ) x . Suppose that d is a ( σ , τ ) -derivation defined on R . In the present paper it is shown that ( i ) if R satisfies [ d ( x ) , x ] σ , τ = 0 , then either d = 0 or R is commutative ( i i ) if I is a nonzero ideal of R such that [ d ( x ) , d ( y ) ] = 0 , for all x , y I , and d commutes with both σ and τ , then either d = 0 or R is commutative. ( i i i ) if I is a nonzero ideal of R such that d ( x y ) = d ( y x ) , for all x , y I , and d commutes with τ , then R is commutative. Finally a related result has been obtain for ( σ , τ ) -derivation....

Generalized Higher Derivations on Lie Ideals of Triangular Algebras

Mohammad AshrafNazia ParveenBilal Ahmad Wani — 2017

Communications in Mathematics

Let 𝔄 = 𝒜 be the triangular algebra consisting of unital algebras 𝒜 and over a commutative ring R with identity 1 and be a unital ( 𝒜 , ) -bimodule. An additive subgroup 𝔏 of 𝔄 is said to be a Lie ideal of 𝔄 if [ 𝔏 , 𝔄 ] 𝔏 . A non-central square closed Lie ideal 𝔏 of 𝔄 is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on 𝔄 , every generalized Jordan triple higher derivation of 𝔏 into 𝔄 is a generalized higher derivation of 𝔏 into 𝔄 .

Nonlinear * -Lie higher derivations of standard operator algebras

Mohammad AshrafShakir AliBilal Ahmad Wani — 2018

Communications in Mathematics

Let be an infinite-dimensional complex Hilbert space and 𝔄  be a standard operator algebra on which is closed under the adjoint operation. It is shown that every nonlinear * -Lie higher derivation 𝒟 = { δ n } n of 𝔄 is automatically an additive higher derivation on 𝔄 . Moreover, 𝒟 = { δ n } n is an inner * -higher derivation.

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