We introduce the notion of n‐homoderivation on a ring ℜ and show that a semiprime ring ℜ must have a nontrivial central ideal if it admits an appropriate n‐homoderivation which is centralizing on some nontrivial one‐sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

The purpose of this paper is to prove the following assertion. Let S be a ring having a nontrivial idempotent element e which satisfies some conditions. We prove that any multiplicative centrally-extended derivation of S is a centrally-extended derivation.

Let be a ring with involution having a nontrivial symmetric idempotent element . If is any appropriate multiplicative generalized reverse CE-derivation of with involution , then under some suitable restrictions on , is centrally-extended additive.

In this paper, we show that for semi-prime rings of two-torsion free and 6-centrally torsion free, given a multiplicative centrally-extended derivation δ and a multiplicative centrally-extended epimorphism ϕ we can find a central ideal K and maps $\mathrm{\Delta },\mathrm{\Phi }:S/K\to S/K$ such as Δ and Φ are multiplicative derivation and multiplicative epimorphism, respectively.

In this article, we examine the commutativity of a ring $\Omega$ endowed with a specific kind of mappings called centrally extended $(\Phi, m)$-homoderivations, where $\Phi$ is a mapping on $\Omega$, and $m$ is an integer. This mapping is a comprehensive kind of the homoderivation, $\Phi- $homoderivation, and $ m $-homoderivation. Besides, we provide some properties of its center.