Ring in Which Every Element is Sum of Two 5-Potent Elements

Abstract

Every element of a ring $R$ is a sum of two commuting 5-potents if and only if $R\cong R_1\times R_2\times R_3\times R_4$, where $R_1/J(R_1)$ is Boolean and $U(R_1)$ is a group of exponent $4$, $R_2$ is a subdirect product of $Z_3$'s, $R_3$ is a subdirect product of $Z_5$'s and $R_4$ is a subdirect product of $Z_{13}$'s. Also, if in a ring $R$ every element is a sum of two 5-potents and a nilpotent that commute with one another then $R\cong R_1\times R_2\times R_3\times R_4$ where $R_1/J(R_1)$ is Boolean and $J(R_1)$ is nil, $R_2\cong R_a\times R_b\times R_c$ where $R_a=0$, $R_c=0$ and $R_b/J(R_b)$ is a subdirect product of rings isomorphic to $Z_3$, $M_2(Z_3)$ or $F_9$ with $J(R_b)$ is nil, $R_3/J(R_3)$ is a subdirect product of $Z_5$'s and $J(R_3)$ is nil, $R_4/J(R_4)$ is a subdirect product of $Z_{13}$'s and $J(R_4)$ is nil.