Automorphisms of completely primary finite rings of characteristic p

Chiteng'a John Chikunji

Displaying similar documents to “Automorphisms of completely primary finite rings of characteristic p”

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.

On near-ring ideals with $(σ, τ)$ -derivation

Öznur Golbaşi, Neşet Aydin (2007)

Archivum Mathematicum

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Let $N$ be a $3$ -prime left near-ring with multiplicative center $Z$ , a $(σ, τ)$ -derivation $D$ on $N$ is defined to be an additive endomorphism satisfying the product rule $D (x y) = τ (x) D (y) + D (x) σ (y)$ for all $x, y \in N$ , where $σ$ and $τ$ are automorphisms of $N$ . A nonempty subset $U$ of $N$ will be called a semigroup right ideal (resp. semigroup left ideal) if $U N \subset U$ (resp. $N U \subset U$ ) and if $U$ is both a semigroup right ideal and a semigroup left ideal, it be called a semigroup ideal. We prove the following results: Let $D$ be a $(σ,$ $τ)$ -derivation...

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...

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.

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...

On feebly nil-clean rings

Marjan Sheibani Abdolyousefi, Neda Pouyan (2024)

Czechoslovak Mathematical Journal

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A ring $R$ is feebly nil-clean if for any $a \in R$ there exist two orthogonal idempotents $e, f \in R$ and a nilpotent $w \in R$ such that $a = e - f + w$ . Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.

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 $f \in k [x, y] ∖ k [x^{p}, y^{p}]$ .

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 $k$ .

Automorphisms of $(λ) / ℐ_{κ}$

Paul Larson, Paul McKenney (2016)

Fundamenta Mathematicae

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We study conditions on automorphisms of Boolean algebras of the form $(λ) / ℐ_{κ}$ (where λ is an uncountable cardinal and $ℐ_{κ}$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $(2^{κ}) / ℐ_{κ ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M A_{ℵ ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial. ...

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 $_{R, n}$ . ...

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...

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.