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

Piotr Jędrzejewicz

Displaying similar documents to “A Characterization of One-Element p-Bases of Rings of Constants”

Avoidance principle and intersection property for a class of rings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with identity. If a ring $R$ is contained in an arbitrary union of rings, then $R$ is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in $R$ , then $R$ contains one of them under various conditions.

Notes on generalizations of Bézout rings

Haitham El Alaoui, Hakima Mouanis (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we give new characterizations of the $P$ - $2$ -Bézout property of trivial ring extensions. Also, we investigate the transfer of this property to homomorphic images and to finite direct products. Our results generate original examples which enrich the current literature with new examples of non- $2$ -Bézout $P$ - $2$ -Bézout rings and examples of non- $P$ -Bézout $P$ - $2$ -Bézout rings.

A generalization of reflexive rings

Mete Burak Çalcı, Huanyin Chen, Sait Halıcıoğlu (2024)

Mathematica Bohemica

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We introduce a class of rings which is a generalization of reflexive rings and $J$ -reversible rings. Let $R$ be a ring with identity and $J (R)$ denote the Jacobson radical of $R$ . A ring $R$ is called $J$ -reflexive if for any $a, b \in R$ , $a R b = 0$ implies $b R a \subseteq J (R)$ . We give some characterizations of a $J$ -reflexive ring. We prove that some results of reflexive rings can be extended to $J$ -reflexive rings for this general setting. We conclude some relations between $J$ -reflexive rings and some related rings. We investigate some extensions...

Left EM rings

Jongwook Baeck (2024)

Czechoslovak Mathematical Journal

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Let $R [x]$ be the polynomial ring over a ring $R$ with unity. A polynomial $f (x) \in R [x]$ is referred to as a left annihilating content polynomial (left ACP) if there exist an element $r \in R$ and a polynomial $g (x) \in R [x]$ such that $f (x) = r g (x)$ and $g (x)$ is not a right zero-divisor polynomial in $R [x]$ . A ring $R$ is referred to as left EM if each polynomial $f (x) \in R [x]$ is a left ACP. We observe the structure of left EM rings with various properties, and study the relationships between the one-sided EM condition and other standard ring theoretic conditions....

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) ...

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 weakened $(α, δ)$ -skew Armendariz rings

Alireza Majdabadi Farahani, Mohammad Maghasedi, Farideh Heydari, Hamidagha Tavallaee (2022)

Mathematica Bohemica

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In this note, for a ring endomorphism $α$ and an $α$ -derivation $δ$ of a ring $R$ , the notion of weakened $(α, δ)$ -skew Armendariz rings is introduced as a generalization of $α$ -rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(α, δ)$ -skew Armendariz ring if and only if $T_{n} (R)$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz if and only if $R [x] / (x^{n})$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz ring for any positive integer $n$ .

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.

Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez (2002)

Studia Mathematica

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In rings $Γ_{M}$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M = {(M_{l})}_{l \geq 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in $Γ_{M}$ such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to $Γ_{M}$ , provided $Γ_{M}$ is stable under derivation. By a further development of the method, we obtain the main result of the paper,...

A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor (2015)

Communications in Mathematics

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In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$ , and let $J^{#}$ denote the set of all elements of $R$ which are nilpotent in $R / J$ . An element $a \in R$ is called provided that there exists an idempotent $e \in R$ such that $a e = e a$ and $a - e$ or $a + e$ is an element of $J^{#}$ . A ring $R$ is said to be in case every element in $R$ is very $J^{#}$ -clean. We prove that every very $J^{#}$ -clean ring is strongly $π$ -rad clean and has stable range one. It is shown that for a...

Sidon basis in polynomial rings over finite fields

Wentang Kuo, Shuntaro Yamagishi (2021)

Czechoslovak Mathematical Journal

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Let $𝔽_{q} [t]$ denote the polynomial ring over $𝔽_{q}$ , the finite field of $q$ elements. Suppose the characteristic of $𝔽_{q}$ is not $2$ or $3$ . We prove that there exist infinitely many $N \in ℕ$ such that the set ${f \in 𝔽_{q} [t] : deg f < N}$ contains a Sidon set which is an additive basis of order $3$ .