Derivations with power central values on Lie ideals in prime rings

Basudeb Dhara; Rajendra K. Sharma

Displaying similar documents to “Derivations with power central values on Lie ideals in prime rings”

Derivations with Engel conditions in prime and semiprime rings

Shuliang Huang (2011)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ , $d$ a derivation of $R$ and $m, n$ fixed positive integers. (i) If ${(d [x, y])}^{m} = {[x, y]}_{n}$ for all $x, y \in I$ , then $R$ is commutative. (ii) If $Char R \neq 2$ and ${[d (x), d (y)]}_{m} = {[x, y]}^{n}$ for all $x, y \in I$ , then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

Generalized derivations in prime rings and Banach algebras

Asma Ali, Basudeb Dhara, Shahoor Khan (2014)

Discussiones Mathematicae - General Algebra and Applications

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Let R be a prime ring with extended centroid C, F a generalized derivation of R and n ≥ 1, m≥ 1 fixed integers. In this paper we study the situations: 1. ${(F (x \circ y))}^{m} = (x \circ y) ⁿ$ for all x,y ∈ I, where I is a nonzero ideal of R; 2. (F(x∘y))ⁿ=(x∘y)ⁿ for all x,y ∈ I, where I is a nonzero right ideal of R. Moreover, we also investigate the situation in semiprime rings and Banach algebras.

An identity with generalized derivations on Lie ideals, right ideals and Banach algebras

Vincenzo de Filippis, Giovanni Scudo, Mohammad S. Tammam El-Sayiad (2012)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring of characteristic different from $2$ , $U$ the Utumi quotient ring of $R$ , $C = Z (U)$ the extended centroid of $R$ , $L$ a non-central Lie ideal of $R$ , $F$ a non-zero generalized derivation of $R$ . Suppose that $[F (u), u] F (u) = 0$ for all $u \in L$ , then one of the following holds: (1) there exists $α \in C$ such that $F (x) = α x$ for all $x \in R$ ; (2) $R$ satisfies the standard identity $s_{4}$ and there exist $a \in U$ and $α \in C$ such that $F (x) = a x + x a + α x$ for all $x \in R$ . We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion...

Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings

Vijay Kumar Bhat (2013)

Czechoslovak Mathematical Journal

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Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $σ$ be an automorphism of $R$ and $δ$ a $σ$ -derivation of $R$ . Then $R$ is said to be an almost $δ$ -divided ring if every minimal prime ideal of $R$ is $δ$ -divided. Let $R$ be a Noetherian ring which is also an algebra over $ℚ$ ( $ℚ$ is the field of rational numbers). Let $σ$ be an automorphism of $R$ such that $R$ is a $σ (*)$ -ring and $δ$ a $σ$ -derivation of $R$ such that $σ (δ (a)) = δ (σ (a))$ for all $a \in R$ . Further,...

On a subset with nilpotent values in a prime ring with derivation

Vincenzo De Filippis (2002)

Bollettino dell'Unione Matematica Italiana

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Let $R$ be a prime ring, with no non-zero nil right ideal, $d$ a non-zero drivation of $R$ , $I$ a non-zero two-sided ideal of $R$ . If, for any $x$ , $y \in I$ , there exists $n = n (x, y) \geq 1$ such that ${(d ([x, y]) - [x, y])}^{n} = 0$ , then $R$ is commutative. As a consequence we extend the result to Lie ideals.