On symmetric bi-derivations acting upon prime ideals in any rings - Sögütcü, Emine Koç; Gölbaşı, Öznur; Dhara, Basudeb (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 512.5Let $R$ be a ring, $P$ be a prime ideal of $R$, $I$ be a nonzero ideal of $R$ such that $P\subsetneq I$, $D_{1}, D_{2}, D_{3}\colon R× R→ R$ 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) $d_1(x)° x\in P,$ (iii) $D_{1}(d_{2}(x),x)\in P,$ (iv) $d_{1}(d_{2}(x))± d_{3}(x)\in P,$ and (v) $d_{1}(x)d_{2}(x)\in P$ for all $x,y\in I.$