When is every matrix over a ring the sum of two tripotents? (Q821020)

The main result of the paper shows the equivalence of the following conditions over an integral domain \(R\): (1) Every matrix in \(M_n(R)\) is the sum of two tripotents for all \(n\in \mathbb{N}\); (2) There exists a positive integer \(n\) such that every matrix in \(M_n(R)\) is the sum of two tripotents; (3) Every matrix in \(M_n(R)\) is the sum of an idempotent and a 5-potent for all \(n\in \mathbb{N}\); (4) There exists a positive integer \(n\) such that every matrix in \(M_n(R)\) is the sum of an idempotent and a 5-potent; (5) \(R\) is isomorphic to \(\mathbb{F}_2\), or \(\mathbb{F}_3\), or \(\mathbb{F}_5\). As applications, the authors prove that if \(R\) is a commutative ring either without any non-zero nilpotent or having 2 as a unit, then the above first four conditions are still equivalent, while the fifth condition is replaced by \(R\) having the identity \(x^5=x\).