Rings in which Every Element is a Sum of Two Tripotents

Abstract

AbstractLet 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 x6 = x4 if and only if R ≅ R1 × R2, where R1/J(21) is Boolean with U(R1) a group of exponent 2 and R2 is zero or a subdirect product of ℤ3’s. (2) Every element of R is either a sum or a difference of two commuting idempotents if and only if R ≅ R1 × R2, where R1/J(R1) is Boolean with J(R1) = 0 or J(R1) = {0, 2} and R2 is zero or a subdirect product of ℤ3’s. (3) Every element of R is a sum of two commuting tripotents if and only if R ≅ R1 × R2 × R3, where R1/J(R1) is Boolean with U(R1) a group of exponent 2, R2 is zero or a subdirect product of ℤ3’s, and R3 is zero or a subdirect product of ℤ5’s.