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

In this article, we define and discuss strongly S3,7 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 S3,7 nil-clean ring if and only if R=R1⊕R2⊕R3⊕R4⊕R5⊕R6, where R1, R2, R3, R4, R5, and R6 are strongly S3,7 nil-clean rings with 2∈NilR1, 3∈NilR2, 5∈NilR3, 7∈NilR4, 13∈NilR5, and 19∈NilR6. The equivalent conditions of strongly S3,7 nil-clean rings in some cases are discussed.