| Summary: | We completely characterize up to an isomorphism those rings whose elements satisfy the equations $ x^4=x $ or $ x^4=-x $ . Specifically, it is proved that a ring is weakly quadratent if, and only if, it is isomorphic to either K, $ \mathbb {Z}_3 $ , $ \mathbb {Z}_7 $ , $ K\times \mathbb {Z}_3 $ or $ K\times \mathbb {Z}_7 $ , where K is a ring which is a subring of a direct product of family of copies of the fields $ \mathbb {Z}_2 $ and $ \mathbb {F}_4 $ . This achievement continues our recent joint investigation in J. Algebra (2015) where we have characterized weakly boolean rings satisfying the equations $ x^2=x $ or $ x^2=-x $ as well as a recent own investigation in Kragujevac J. Math. (2019) where we have characterized weakly tripotent rings satisfying the equations $ x^3=x $ or $ x^3=-x $ .
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