On the Casas-Alvero’ s Conjecture Su, Jiaxuan
Journal of physics. Conference series,
06/2021, Letnik:
1955, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Abstract
As we all know, every derivation of a polynomial that is a nth degree of another polynomial has a common factor with this polynomial. Then, Eduardo Casas-Alvero came up with a conjecture ...that think opposite with the axiom we just talked before. That is to proof this polynomial is a nth degree of another polynomial if every derivation of this polynomial has a common factor with this polynomial. We are going to use different method to attack this conjecture and prove it when n is small.
Let R be a 2-torsion free commutative ring with identity, A,B be unital algebras over R and M be a unital (A,B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let T=AM0B ...be the triangular algebra consisting of A,B and M, and let d be an R-linear mapping from T into itself. Suppose that A and B have only trivial idempotents. Then the following statements are equivalent: (1) d is a Jordan (α,β)-derivation on T; (2) d is a Jordan triple (α,β)-derivation on T; (3) d is an (α,β)-derivation on T. Furthermore, a generalized version of this result is also given. We characterize the actions of automorphisms and skew derivations on the triangular algebra T. The structure of continuous (α,β)-derivations of triangular Banach algebras and that of generalized Jordan (α,β)-derivations of upper triangular matrix algebras are described.
Let
R
be a 2-torsion free commutative ring with unity and X be a locally finite preordered set. We prove in this paper that every nonlinear Jordan higher derivation on the incidence algebra
I
(
X
,
R
...)
is an additive higher derivation, provided that all connected components of X are nontrivial.
Using the first cohomology from the mirror Heisenberg–Virasoro algebra to the twisted Heisenberg algebra (as the mirror Heisenberg–Virasoro algebra module), in this paper, we determined the ...derivations on the mirror Heisenberg–Virasoro algebra. Based on this result, we proved that any two-local derivation on the mirror Heisenberg–Virasoro algebra is a derivation. All half-derivations are described, and as corollaries, we have descriptions of transposed Poisson structures and local (two-local) half-derivations on the mirror Heisenberg–Virasoro algebra.
σ-derivations on generalized matrix algebras Jabeen, Aisha; Ashraf, Mohammad; Ahmad, Musheer
Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică,
07/2020, Letnik:
28, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Let be a commutative ring with unity, 𝒜, be -algebras, be (𝒜, )-bimodule and 𝒩 be (, 𝒜)-bimodule. The -algebra 𝒢 = 𝒢(𝒜, , 𝒩, ) is a generalized matrix algebra defined by ...the Morita context (𝒜, , , 𝒩, ξ
, Ω
). In this article, we study Jordan
derivations on generalized matrix algebras.
In this paper, we introduce a new class of derivations that generalizes skew derivations and semi-derivations, and we call it skew semi-derivation.
Furthermore, we present a study of the conditions ...under which this type of multiplicative derivation becomes additive.
Let
be a triangular algebra. We show that under suitable assumptions every generalized Lie n-derivation
associated with a linear map
is of the form
where
and Δ is a Lie n-derivation of
We solve this ...problem using commuting and centralizing maps. We also prove that under certain mild conditions any centralizing map on a triangular algebra is commuting. As an application, we give a description of generalized Lie n-derivations on classical examples of triangular algebras: upper triangular matrix algebras and nest algebras.
Let
be a unital algebra with a nontrivial idempotent
over a unital commutative ring
. We show that under suitable assumptions, every Lie triple derivation
on
is of the form
, where
is a derivation of
...,
is a singular Jordan derivation of
and
is a linear mapping from
to its centre
that vanishes on
. As an application, we characterize Lie triple derivations and Lie derivations on triangular algebras and on matrix algebras.
In 1977, Colville, Davis, and Keimel Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375 proved that a positive derivation on an Archimedean f-algebra A has its range in ...the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras.
PDF
