Group rings that are weakly quadratent

A ring $ R $ is said to be weakly quadratent if every element $ x\in R $ satisfies $ x^{4} = x $ or $ x^{4} = -x $ ^[8]. In this paper, we determine, for an associative and unital ring $ R $ and an arbitrary group $ G $, exactly when the group ring $ RG $ is weakly quadratent. Specifically, we prove that $ RG $ is weakly quadratent if and only if either $ G $ is trivial and $ R $ is weakly quadratent or $ G $ is a non-trivial elementary abelian $ 3 $-group and $ R $ is a quadratent ring, that is, a subring of a direct product of copies of finite fields $ \mathbb{F}_2 $ and $ \mathbb{F}_4 $; in the latter case, $ RG $ is in fact quadratent. Besides, we establish connections between our results and the von Neumann regular group rings and, by means of explicit examples, demonstrate how far semiprimeness is from guaranteeing the desired properties. Our work extends the existing line of research on weakly Boolean, weakly tripotent, and weakly quadratent rings, as well as their analogues in group rings. Finally, we present several open research problems that may lead to further developments in this area.