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∈Nil<mfenced open="(" close=")" separators="|">R1</mfenced> , 3∈Nil<mfenced open="(" close=")" separators="|">R2</mfenced> , 5∈Nil<mfenced open="(" close=")" separators="|">R3</mfenced> , 7∈Nil<mfenced open="(" close=")" separators="|">R4</mfenced> , 13∈Nil<mfenced open="(" close=")" separators="|">R5</mfenced> , and 19∈Nil<mfenced open="(" close=")" separators="|">R6</mfenced> . The equivalent conditions of strongly S3,7 nil-clean rings in some cases are discussed.