STRUCTURE OF A RING IN WHICH EVERY ELEMENT IS SUM OF 3, 4 OR 5 COMMUTING TRIPOTENTS

In this paper we show if $R$ be a ring in which every element is sum of three commuting tripotents then for every $k\in R$ we have $(k-3)(k-2)^2(k-1)^2k^2(k+1)^2(k+2)^2(k+3)=0$, if every element of $R$ is sum of four commuting tripotents then for every $k\in R$ we have $(k-4)(k-3)(k-2)^2(k-1)^2k^4(k+1)^2(k+2)^2(k+3)(k+4)=0$, if every element of $R$ is sum of five commuting tripotents then for every $k\in R$ we have $(k-5)(k-4)(k-3)^2(k-2)^3(k-1)^3k^4(k+1)^3(k+2)^3(k+3)^2(k+4)(k+5)=0$. Then we discuss the properties of these type of ring. Finally we find the general structure of a ring in which every element is sum of $n$ commuting tripotents and discuss the properties of it.