arXiv subjects
Publications and source records attributed to L. Oukhtite.
Let $R$ be a 2-torsion free $σ$-prime ring, $U$ a nonzero square closed $σ$-Lie ideal of $R$ and let $d$ be a derivation of $R$. In this paper it is shown that: 1) If $d$ is centralizing on $U$, then $d = 0$ or $U \subseteq Z(R)$. 2) If either $d([x, y]) = 0$ for all $x, y \in U$, or $[d(x), d(y)] = 0$ for all $x, y \in U$ and $d$ commutes with $σ$ on $U$, then $d = 0$ or $U \subseteq Z(R)$.
Let $R$ be a 2-torsion free semiprime ring and $U$ a nonzero square closed Lie ideal of $R$. In this paper it is shown that if $f$ is either an endomorphism or an antihomomorphism of $R$ such that $f(U)=U,$ then $f$ is strong commutativity preserving on $U$ if and only if $f$ is centralizing on $U.$
Let $R$ be a 2-torsion free $σ$-prime ring. It is shown here that if $U\not\subset Z(R)$ is a $σ$-Lie ideal of $R$ and $a, b$ in $R$ such that $aUb=σ(a)Ub=0,$ then either $a=0$ or $b=0.$ This result is then applied to study the relationship between the structure of $R$ and certain automorphisms on $R$. To end this paper, we describe additive maps $d: R \longrightarrow R$ such that $d(u^2) = 2ud(u)$ where $u\in U,$ a nonzero $σ$-square closed Lie ideal of $R.$