Displaying similar documents to “Posner's second theorem and annihilator conditions with generalized skew 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 ( σ , τ ) -derivations in prime rings

Mohammad Ashraf, Nadeem-ur-Rehman (2002)

Archivum Mathematicum

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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...

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 ] .

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.

A Characterization of One-Element p-Bases of Rings of Constants

Piotr Jędrzejewicz (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in K [ x p , . . . , x p ] . We prove that K [ x p , . . . , x p , f ] is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

A note on the kernels of higher derivations

Jiantao Li, Xiankun Du (2013)

Czechoslovak Mathematical Journal

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Let k k ' be a field extension. We give relations between the kernels of higher derivations on k [ X ] and k ' [ X ] , where k [ X ] : = k [ x 1 , , x n ] denotes the polynomial ring in n variables over the field k . More precisely, let D = { D n } n = 0 a higher k -derivation on k [ X ] and D ' = { D n ' } n = 0 a higher k ' -derivation on k ' [ X ] such that D m ' ( x i ) = D m ( x i ) for all m 0 and i = 1 , 2 , , n . Then (1) k [ X ] D = k if and only if k ' [ X ] D ' = k ' ; (2) k [ X ] D is a finitely generated k -algebra if and only if k ' [ X ] D ' is a finitely generated k ' -algebra. Furthermore, we also show that the kernel k [ X ] D of a higher derivation D of k [ X ] can be generated...

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: ...