Displaying similar documents to “A note on the kernels of higher derivations”

A note on linear derivations

Amit Patra (2024)

Czechoslovak Mathematical Journal

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At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer w > 1 such that the polynomial ring in n variables is w -differentially simple, all w derivations are nonsimple and the w derivations set contains a linear derivation.

On skew derivations as homomorphisms or anti-homomorphisms

Mohd Arif Raza, Nadeem ur Rehman, Shuliang Huang (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a prime ring with center Z and I be a nonzero ideal of R . In this manuscript, we investigate the action of skew derivation ( δ , ϕ ) of R which acts as a homomorphism or an anti-homomorphism on I . Moreover, we provide an example for semiprime case.

Generalized derivations on Lie ideals in prime rings

Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal (2015)

Czechoslovak Mathematical Journal

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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: ...

Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let R be a prime ring of characteristic different from 2, Q r its right Martindale quotient ring and C its extended centroid. Suppose that F , G are generalized skew derivations of R with the same associated automorphism α , and p ( x 1 , ... , x n ) is a non-central polynomial over C such that [ F ( x ) , α ( y ) ] = G ( [ x , y ] ) for all x , y { p ( r 1 , ... , r n ) : r 1 , ... , r n R } . Then there exists λ C such that F ( x ) = G ( x ) = λ α ( x ) for all x R .

On rings of constants of derivations in two variables in positive characteristic

Piotr Jędrzejewicz (2006)

Colloquium Mathematicae

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Let k be a field of chracteristic p > 0. We describe all derivations of the polynomial algebra k[x,y], homogeneous with respect to a given weight vector, in particular all monomial derivations, with the ring of constants of the form k [ x p , y p , f ] , where f k [ x , y ] k [ x p , y p ] .

Generalized derivations with power values on rings and Banach algebras

Abderrahman Hermas, Abdellah Mamouni, Lahcen Oukhtite (2024)

Mathematica Bohemica

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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

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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.

On generalized derivations of partially ordered sets

Ahmed Y. Abdelwanis, Abdelkarim Boua (2019)

Communications in Mathematics

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Let P be a poset and d be a derivation on P . In this research, the notion of generalized d -derivation on partially ordered sets is presented and studied. Several characterization theorems on generalized d -derivations are introduced. The properties of the fixed points based on the generalized d -derivations are examined. The properties of ideals and operations related with generalized d -derivations are studied.