Rings in which Every Element is a Sum of Two Tripotents

Let R be a ring. The following results are proved. (1) Every element of R is a sum of an idempotent and a tripotent that commute if and only if R has the identity x(6) = x(4) if and only if R congruent to R-1 x R-2, where R-1/J(R-1) is Boolean with U(R-1) a group of exponent 2 and R-2 is zero or a subdirect product of Z(3)'s. (2) Every element of R is either a sum or a difference of two commuting idempotents if and only if R congruent to R-1 x R-2, where R-1/J(R-1) is Boolean with J(R-1) = 0 or J(R-1) = {0, 2} and R-2 is zero or a subdirect product of Z(3)'s. (3) Every element of R is a sum of two commuting tripotents if and only if R congruent to R-1 x R-2 x R-3, where R-1/J(R-1) is Boolean with U(R-1) a group of exponent 2, R-2 is zero or a subdirect product of Z(3)'s, and R-3 is zero or a subdirect product of Z(5)'s.