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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.

Derivations with power central values on Lie ideals in prime rings

Basudeb Dhara, Rajendra K. Sharma (2008)

Czechoslovak Mathematical Journal

Let $R$ be a prime ring of char $R \neq 2$ with a nonzero derivation $d$ and let $U$ be its noncentral Lie ideal. If for some fixed integers $n_{1} \geq 0, n_{2} \geq 0, n_{3} \geq 0$ , ${(u^{n_{1}} [d (u), u] u^{n_{2}})}^{n_{3}} \in Z (R)$ for all $u \in U$ , then $R$ satisfies $S_{4}$ , the standard identity in four variables.

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Decomposition of rings under the circle operation.

Defect And Radicals Of Δ-Endomorphism Near-Rings

Dependent elements of derivations on semiprime rings.

Der Wedderburnsche Hauptsatz für alternative Tripelsysteme und Paare.

Derivation-like mappings for normal elements in prime rings with involution.

Derivations and multilinear polynomials

Derivations in rings with involution

Derivations of higher order in semiprime rings.

Derivations on noncommutative Banach algebras

Derivations with Engel conditions in prime and semiprime rings

Derivations with power central values on Lie ideals in prime rings

Derivazioni in anelli primi

Die absteigende Loewy-Länge von Endomorphismenringen.

Die Ringe mit einem Unterring, der mit allen Unterringen vergleichbar ist.

Direct sums of $J$ -rings and radical rings.

Dreiecks-Darstellung von Endomorphismenringen injektiver Moduln.

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