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Derivations with Engel conditions in prime and semiprime rings

Shuliang Huang (2011)

Czechoslovak Mathematical Journal

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 [ x , y ] ) m = [ x , y ] n for all x , y I , then R is commutative. (ii) If Char R 2 and [ d ( x ) , d ( y ) ] m = [ x , y ] n for all x , y I , then R is commutative. Moreover, we also examine the case when R is a semiprime ring.

Derivations with power central values on Lie ideals in prime rings

Basudeb Dhara, Rajendra K. Sharma (2008)

Czechoslovak Mathematical Journal

Let R be a prime ring of char R 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n 1 0 , n 2 0 , n 3 0 , ( u n 1 [ d ( u ) , u ] u n 2 ) n 3 Z ( R ) for all u U , then R satisfies S 4 , the standard identity in four variables.

Essential Cover and Closure

Andruszkiewicz, R. (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 16N80, 16S70, 16D25, 13G05.We construct some new examples showing that Heyman and Roos construction of the essential closure in the class of associative rings can terminate at any finite or the first infinite ordinal.

Free actions on semiprime rings

Muhammad Anwar Chaudhry, Mohammad S. Samman (2008)

Mathematica Bohemica

We identify some situations where mappings related to left centralizers, derivations and generalized ( α , β ) -derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation T , of a semiprime ring R the mapping ψ R R defined by ψ ( x ) = T ( x ) x - x T ( x ) for all x R is a free action. We also show that for a generalized ( α , β ) -derivation F of a semiprime ring R , with associated ( α , β ) -derivation d , a dependent element a of F is also a dependent element of α + d . Furthermore, we prove that for a centralizer f and...

Generalized derivations on Lie ideals in prime rings

Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal (2015)

Czechoslovak Mathematical Journal

Let R be a prime ring with its Utumi ring of quotients U and extended centroid C . Suppose that F is a generalized derivation of R and L is a noncentral Lie ideal of R such that F ( u ) [ F ( u ) , u ] n = 0 for all u L , where n 1 is a fixed integer. Then one of the following holds: ...

Generalized derivations with power values on rings and Banach algebras

Abderrahman Hermas, Abdellah Mamouni, Lahcen Oukhtite (2024)

Mathematica Bohemica

Let R be a prime ring and I a nonzero ideal of R . The purpose of this paper is to classify generalized derivations of R satisfying some algebraic identities with power values on I . More precisely, we consider two generalized derivations F and H of R satisfying one of the following identities: ...

Generalized reverse derivations and commutativity of prime rings

Shuliang Huang (2019)

Communications in Mathematics

Let R be a prime ring with center Z ( R ) and I a nonzero right ideal of R . Suppose that R admits a generalized reverse derivation ( F , d ) such that d ( Z ( R ) ) 0 . In the present paper, we shall prove that if one of the following conditions holds: (i) F ( x y ) ± x y Z ( R ) , (ii) F ( [ x , y ] ) ± [ F ( x ) , y ] Z ( R ) , (iii) F ( [ x , y ] ) ± [ F ( x ) , F ( y ) ] Z ( R ) , (iv) F ( x y ) ± F ( x ) F ( y ) Z ( R ) , (v) [ F ( x ) , y ] ± [ x , F ( y ) ] Z ( R ) , (vi) F ( x ) y ± x F ( y ) Z ( R ) for all x , y I , then R is commutative.

Jordan ideals and derivations in prime near-rings

Abdelkarim Boua, Lahcen Oukhtite, Abderrahmane Raji (2014)

Commentationes Mathematicae Universitatis Carolinae

In this paper we investigate 3 -prime near-rings with derivations satisfying certain differential identities on Jordan ideals, and we provide examples to show that the assumed restrictions cannot be relaxed.

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