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On generalized derivations of partially ordered sets

Ahmed Y. Abdelwanis, Abdelkarim Boua (2019)

Communications in Mathematics

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.

On local derivations in the Kadison sense

Andrzej Nowicki (2001)

Colloquium Mathematicae

Let k be a field. We prove that any polynomial ring over k is a Kadison algebra if and only if k is infinite. Moreover, we present some new examples of Kadison algebras and examples of algebras which are not Kadison algebras.

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

Piotr Jędrzejewicz (2006)

Colloquium Mathematicae

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 the generalized vanishing conjecture

Zhenzhen Feng, Xiaosong Sun (2019)

Czechoslovak Mathematical Journal

We show that the GVC (generalized vanishing conjecture) holds for the differential operator Λ = ( x - Φ ( y ) ) y and all polynomials P ( x , y ) , where Φ ( t ) is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

On the ring of constants for derivations of power series rings in two variables

Leonid Makar-Limanov, Andrzej Nowicki (2001)

Colloquium Mathematicae

Let k[[x,y]] be the formal power series ring in two variables over a field k of characteristic zero and let d be a nonzero derivation of k[[x,y]]. We prove that if Ker(d) ≠ k then Ker(d) = Ker(δ), where δ is a jacobian derivation of k[[x,y]]. Moreover, Ker(d) is of the form k[[h]] for some h ∈ k[[x,y]].

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