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

Piotr Jędrzejewicz

Displaying similar documents to “On rings of constants of derivations in two variables in positive characteristic”

Linear derivations with rings of constants generated by linear forms

Piotr Jędrzejewicz (2008)

Colloquium Mathematicae

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Let k be a field. We describe all linear derivations d of the polynomial algebra k[x₁,...,xₘ] such that the algebra of constants with respect to d is generated by linear forms: (a) over k in the case of char k = 0, (b) over $k [x ₁^{p}, . . ., x ₘ^{p}]$ in the case of char k = p > 0.

Posner's second theorem and annihilator conditions with generalized skew derivations

Vincenzo De Filippis, Feng Wei (2012)

Colloquium Mathematicae

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Let be a prime ring of characteristic different from 2, $_{r}$ be its right Martindale quotient ring and be its extended centroid. Suppose that is a non-zero generalized skew derivation of and f(x₁,..., xₙ) is a non-central multilinear polynomial over with n non-commuting variables. If there exists a non-zero element a of such that a[ (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ , then one of the following holds: (a) there exists λ ∈ such that (x) = λx for all x ∈ ; (b) there...

About G-rings

Najib Mahdou (2017)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if $R \subseteq T$ is a ring extension such that $m T \subseteq R$ for some regular element $m$ of $T$ , then $T$ is a G-ring if and only if so is $R$ . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

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 $u \in L$ , where $n \geq 1$ is a fixed integer. Then one of the following holds: (1) ...

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.

Skew inverse power series rings over a ring with projective socle

Kamal Paykan (2017)

Czechoslovak Mathematical Journal

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A ring $R$ is called a right $PS$ -ring if its socle, $Soc (R_{R})$ , is projective. Nicholson and Watters have shown that if $R$ is a right $PS$ -ring, then so are the polynomial ring $R [x]$ and power series ring $R [[x]]$ . In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R [[x^{- 1}; α, δ]]$ and the skew polynomial ring $R [x; α, δ]$ , where $R$ is an associative ring equipped with an automorphism $α$ and an $α$ -derivation $δ$ . Our results extend and unify many existing...

On some noetherian rings of $C^{\infty}$ germs on a real closed field

Abdelhafed Elkhadiri (2011)

Annales Polonici Mathematici

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Let R be a real closed field, and denote by $_{R, n}$ the ring of germs, at the origin of Rⁿ, of $C^{\infty}$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring $_{R, n} \subset_{R, n}$ with some natural properties. We prove that, for each n ∈ ℕ, $_{R, n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then $_{ℝ, n} = ₙ$ , where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring $_{R, n}$ . ...

On matrix Lie rings over a commutative ring that contain the special linear Lie ring

Evgenii L. Bashkirov, Esra Pekönür (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $K$ be an associative and commutative ring with $1$ , $k$ a subring of $K$ such that $1 \in k$ , $n \geq 2$ an integer. The paper describes subrings of the general linear Lie ring $g l_{n} (K)$ that contain the Lie ring of all traceless matrices over $k$ .

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 $x, y \in R$ , 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 $x, y \in I$ , 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 $x, y \in I$ , and $d$ commutes with $τ$ , then $R$ is commutative. Finally a related result has been obtain...

A note on the kernels of higher derivations

Jiantao Li, Xiankun Du (2013)

Czechoslovak Mathematical Journal

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Let $k \subseteq k^{'}$ be a field extension. We give relations between the kernels of higher derivations on $k [X]$ and $k^{'} [X]$ , where $k [X] : = k [x_{1}, \dots, x_{n}]$ denotes the polynomial ring in $n$ variables over the field $k$ . More precisely, let $D = {D_{n}}_{n = 0}^{\infty}$ a higher $k$ -derivation on $k [X]$ and $D^{'} = {D_{n}^{'}}_{n = 0}^{\infty}$ a higher $k^{'}$ -derivation on $k^{'} [X]$ such that $D_{m}^{'} (x_{i}) = D_{m} (x_{i})$ for all $m \geq 0$ and $i = 1, 2, \dots, n$ . 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...

An example of a simple derivation in two variables

Andrzej Nowicki (2008)

Colloquium Mathematicae

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Let k be a field of characteristic zero. We prove that the derivation $D = \partial / \partial x + (y^{s} + p x) (\partial / \partial y)$ , where s ≥ 2, 0 ≠ p ∈ k, of the polynomial ring k[x,y] is simple.

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 $x, y \in {p (r_{1}, ..., r_{n}) : r_{1}, ..., r_{n} \in R}$ . Then there exists $λ \in C$ such that $F (x) = G (x) = λ α (x)$ for all $x \in R$ .

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 $K [x ₁^{p}, . . ., x ₙ^{p}]$ . We prove that $K [x ₁^{p}, . . ., x ₙ^{p}, f]$ 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.

Symmetric and reversible properties of bi-amalgamated rings

Antonysamy Aruldoss, Chelliah Selvaraj (2024)

Czechoslovak Mathematical Journal

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Let $f : A \to B$ and $g : A \to C$ be two ring homomorphisms and let $K$ and $K^{'}$ be two ideals of $B$ and $C$ , respectively, such that $f^{- 1} (K) = g^{- 1} (K^{'})$ . We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A ⋈^{f, g} (K, K^{'})$ of $A$ with $(B, C)$ along $(K, K^{'})$ with respect to $(f, g)$ .