Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta ,\varphi )$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.

Let $𝒜$ be a noncommutative prime ring equipped with an involution ‘$*$’, and let ${𝒬}_{ms}\left(𝒜\right)$ be the maximal symmetric ring of quotients of $𝒜$. Consider the additive maps $\mathscr{H}$ and $𝒯:𝒜\to {𝒬}_{ms}\left(𝒜\right)$. We prove the following under some inevitable torsion restrictions. (a) If $m$ and $n$ are fixed positive integers such that $(m+n)𝒯\left(a^2\right)=m𝒯\left(a\right)a^{*}+na𝒯\left(a\right)$ for all $a\in 𝒜$ and $(m+n)\mathscr{H}\left(a^2\right)=m\mathscr{H}\left(a\right)a^{*}+na𝒯\left(a\right)$ for all $a\in 𝒜$, then $\mathscr{H}=0$. (b) If $𝒯\left(aba\right)=a𝒯\left(b\right)a^{*}$ for all $a,b\in 𝒜$, then $𝒯=0$. Furthermore, we characterize Jordan left $\tau$-centralizers in semiprime rings admitting an anti-automorphism $\tau$. As applications, we find the structure of...

Let $R$ be a 2-torsion free prime ring and let $\sigma ,\tau$ be automorphisms of $R$. For any $x,y\in R$, set ${[x,y]}_{\sigma ,\tau }=x\sigma \left(y\right)-\tau \left(y\right)x$. Suppose that $d$ is a $(\sigma ,\tau )$-derivation defined on $R$. In the present paper it is shown that $\left(i\right)$ if $R$ satisfies ${[d\left(x\right),x]}_{\sigma ,\tau }=0$, then either $d=0$ or $R$ is commutative $\left(ii\right)$ if $I$ is a nonzero ideal of $R$ such that $\left[d\right(x),d(y\left)\right]=0$, for all $x,y\in I$, and $d$ commutes with both $\sigma$ and $\tau$, then either $d=0$ or $R$ is commutative. $\left(iii\right)$ if $I$ is a nonzero ideal of $R$ such that $d\left(xy\right)=d\left(yx\right)$, for all $x,y\in I$, and $d$ commutes with $\tau$, then $R$ is commutative. Finally a related result has been obtain for $(\sigma ,\tau )$-derivation....

Let be a prime ring of characteristic different from 2, ${}_r$ be its right Martindale quotient ring and be its extended centroid. Suppose that is a non-zero generalized skew derivation of and f(x₁,..., xₙ) is a non-central multilinear polynomial over with n non-commuting variables. If there exists a non-zero element a of such that a[ (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ , then one of the following holds: (a) there exists λ ∈ such that (x) = λx for all x ∈ ; (b) there exist $q{\in }_r$ and...