Rings in Which Every Element Is a Sum of a Nilpotent and Three 7-Potents

In this article, we define and discuss strongly $S_{3,7}$ nil-clean rings: every element in a ring is the sum of a nilpotent and three 7-potents that commute with each other. We use the properties of nilpotent and 7-potent to conduct in-depth research and a large number of calculations and obtain a nilpotent formula for the constant $a$. Furthermore, we prove that a ring $R$ is a strongly $S_{3,7}$ nil-clean ring if and only if $R=R_1\oplus R_2\oplus R_3\oplus R_4\oplus R_5\oplus R_6$, where $R_1$, $R_2$, $R_3$, $R_4$, $R_5$, and $R_6$ are strongly $S_{3,7}$ nil-clean rings with $2\in \text{Nil}\left(R_1\right)$, $3\in \text{Nil}\left(R_2\right)$, $5\in \text{Nil}\left(R_3\right)$, $7\in \text{Nil}\left(R_4\right)$, $13\in \text{Nil}\left(R_5\right)$, and $19\in \text{Nil}\left(R_6\right)$. The equivalent conditions of strongly $S_{3,7}$ nil-clean rings in some cases are discussed.