Rings in which Every Element is A Sum of a Nilpotent And Three 5-Potents

Abstract In this article, the concept of strongly S 3,5 nil-clean rings is proposed, where each element in the ring is a sum of a nilpotent and three 5-potents that commute with one another. With the use of the 5-potent and nilpotent characteristics, a nilpotent formula about constant a is obtained through in-depth study and calculation. Furthermore, the structure of such rings is characterized, and the direct sum structure is determined. It is proved that a ring R is a strongly S 3,5 nil-clean ring if and only if R = R 1 ⊕ R 2 ⊕ R 3 ⊕ R 4 ⊕ R 5 ⊕ R 6 , where R 1 , R 2 , R 3 , R 4 , R 5 , R 6 are strongly S 3,5 nil-clean rings with 2 ∈ Nil ( R 1 ), 3 ∈ Nil ( R 2 ), 5 ∈ Nil ( R 3 ), 7 ∈ Nil ( R 4 ), 13 ∈ Nil ( R 5 ) and 17 ∈ Nil ( R 6 ). Finally, equivalent conditions to such rings are provided under certain constraints.