## Rings in which Every Element is a Sum of Two Tripotents

CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, vol.59, no.3, pp.661-672, 2016 (SCI-Expanded)

• Publication Type: Article / Article
• Volume: 59 Issue: 3
• Publication Date: 2016
• Doi Number: 10.4153/cmb-2016-009-0
• Journal Name:
• Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
• Page Numbers: pp.661-672
• Keywords: idempotent, tripotent, Boolean ring, polynomial identity x(3) = x, polynomial identity x(6) = x(4), polynomial identity x(8) = x(4), NIL-CLEAN RINGS
• Gazi University Affiliated: No

#### Abstract

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.