On symmetric bi-derivations acting upon prime ideals in any rings

Abstract

UDC 512.5

Let $R$ be a ring, $P$ be a prime ideal of $R$, $I$ be a nonzero ideal of $R$ such that $P\subsetneq I$, $D_{1},  D_{2}, D_{3}\colon R\times R\rightarrow R$ be symmetric bi-derivations, and $d_{1},\ d_{2},$ and $d_{3}$ be the traces of $D_{1},D_{2},$ and $D_{3}$ respectively. We study the following conditions:  (i) $[d_1(x),x]\in P,$ (ii) $d_1(x)\circ x\in P,$ (iii) $D_{1}(d_{2}(x),x)\in P,$ (iv) $d_{1}(d_{2}(x))\pm d_{3}(x)\in P,$ and (v) $d_{1}(x)d_{2}(x)\in P$ for all $x,y\in I.$