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 ≅ R 1 × R 2 , where R 1 / J ( 2 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 ≅ R 1 × 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 ≅ R 1 × R 2 × 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.