Let $R$ be a prime ring, with no non-zero nil right ideal, $d$ a non-zero drivation of $R$, $I$ a non-zero two-sided ideal of $R$. If, for any $x$, $y\in I$, there exists $n=n\left(x,y\right)\ge 1$ such that ${\left(d\left(\left[x,y\right]\right)-\left[x,y\right]\right)}^n=0$, then $R$ is commutative. As a consequence we extend the result to Lie ideals.

Let $𝒦$ be a semiprime ring and $T:𝒦\to 𝒦$ an additive mapping such that $T\left(x^2\right)=T\left(x\right)x$ holds for all $x\in 𝒦$. Then $T$ is a left centralizer of $𝒦$. It is also proved that Jordan centralizers and centralizers of $𝒦$ coincide.

Let $R$ be a $2$-torsion free $*$-prime ring, $d$ a derivation which commutes with $*$ and $J$ a $*$-Jordan ideal and a subring of $R$. In this paper, it is shown that if either $d$ acts as a homomorphism or as an anti-homomorphism on $J$, then $d=0$ or $J\subseteq Z\left(R\right)$. Furthermore, an example is given to demonstrate that the $*$-primeness hypothesis is not superfluous.

Let $R$ be a 2-torsion free prime ring and let $U$ be a Lie ideal of $R$ such that $u^2\in U$ for all $u\in U$. In the present paper it is shown that if $d$ is an additive mappings of $R$ into itself satisfying $d\left(u^2\right)=2ud\left(u\right)$ for all $u\in U$, then $d\left(uv\right)=ud\left(v\right)+vd\left(u\right)$ for all $u,v\in U$.

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x ↦ xa² for non-zero a, in place of requiring that positive elements have a positive product. Our aim in this work is to study this type of ordering in the case of a division ring. We show that it actually behaves just as in the commutative...

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under $x\to x{a}^2$ for nonzero $a$, instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative...

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...