University of Guilan Journal of Algebra and Related Topics 2345-3931 Articles in Press 2024 12 04 Ring in which every element is sum of two 6-potent elements 8204 10.22124/jart.2024.26096.1605 EN K. N. Deka Department of‎ ‎Mathematics‎,‎ Gauhati University‎,‎ Guwahati‎, ‎India H. K. Saikia Department of‎ ‎Mathematics‎,‎ Gauhati University‎,‎ Guwahati‎, ‎India Journal Article 2023 11 23 ‎In this paper, we prove the following results‎. ‎Every element of a ring $R$ is a sum of two commuting $6-$potent elements if and only if $R$ is isomorphic to $R_1\times R_2\times R_3$‎, ‎where $R_1$ is isomorphic to a subdirect product of $Z_2$'s‎, ‎$R_2$ is isomorphic to a subdirect product of $Z_3$'s and $R_3$ is isomorphic to a subdirect product of $Z_{11}$'s‎. ‎Also, if every element of a ring $R$ is the sum of two 6-potent and one nilpotent all commute with each other, then $R$ is isomorphic to $R_1\times R_2\times R_3$‎, ‎where $J(R_1)$ is nil and $R_1/J(R_1)$ is a subdirect product of rings isomorphic to either of the rings $Z_2,F_4,M_2(F_2)$ and $M_2(F_4)$‎ , ‎$a^{81}-a$ is nilpotent for every $a\in R_2$‎ , ‎$J(R_3)$ is nil and $R_3/J(R_3)$ is a subdirect product of $Z_{11}$'s‎. https://jart.guilan.ac.ir/article_8204_3fe73bdefa87d015d975c979c34d550c.pdf