Weakly quadratent rings

• Main Author: Peter V. Danchev
• Format: Article
• Language: English
• Published: Taylor & Francis Group 2019-12-01
• Series: Journal of Taibah University for Science
• Subjects: fields; equations; rings
• Online Access: http://dx.doi.org/10.1080/16583655.2018.1545559

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 $ .