On derivations and semiprime ideal of rings

TL;DR

The paper addresses the problem of deriving constraints in rings when a nonzero ideal $I$ contains a semiprime ideal $T$ and a derivation $d$ satisfies a $T$-valued differential identity. It develops a direct approach to deduce $[d(x),x] ∈ T$ under identities such as $[[d(x),x],x] ∈ T$ (with $char(R/T) ≠ 2$) and then proves that $d$ is a $T$-commuting map on $I$, from which several corollaries follow. These include P-variant Posner-type results for prime ideals: if $P$ is prime and $[[d(x),x],x] ∈ P$ for all $x$, then either $d(R) ⊆ P$ or $R/P$ is commutative; similarly for $[d(x),x]d(x) ∈ T$ or $[d(x),x]x ∈ T$. The work also extends Cheng's differential-identity theorem to semiprime rings, yielding a broader family of semiprime-based commutativity results.

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.

Theorems & Definitions (19)

• Theorem 2.1
• proof
• Corollary 2.2
• Corollary 2.3
• Corollary 2.4
• proof
• Theorem 2.5
• proof
• Corollary 2.6
• Corollary 2.7