Semiprime-ideal variants of some theorems on derivations

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 $K.$ In this article, we show that if $d$ satisfies some $T$-valued identities that it is $T$-commuting. Consequently, we provide semi-prime ideal variants of some known results on derivations.