On derivations and semiprime ideal of rings

TL;DR

This paper investigates conditions under which derivations on rings are $T$-commuting, especially when certain $T$-valued identities are imposed, and extends known results to semiprime ideals.

Contribution

It introduces new conditions for derivations to be $T$-commuting in rings with semiprime ideals and generalizes existing results to this context.

Findings

Derivations satisfying specific $T$-valued identities are $T$-commuting.

Provides semiprime ideal variants of known derivation results.

Establishes conditions linking derivations and semiprime ideals in rings.

Abstract

Let $R$ be an associative ring with a nonzero ideal $I$ and a semiprime ideal $T$ such that $T\subsetneq I.$ Let $K$ be a nonempty subset of $R$ and $d:R\to R$ be a derivation of $R$, if $[d(x),x]\in T$ for all $x\in K,$ then $d$ is said to be a $T$-commuting derivation on $K.$ We show that if some specific $T$-valued differential identities are imposed on $I$, then d is $T$-commuting. Moreover, we provide semiprime ideal variant of some known results on derivations.