Rings in which every element is the sum of two idempotents

Abstract

RINGS IN WHICH EVERY ELEMENT IS THE SUM OF TWO IDEMPOTENTSYASUYUKI HIRANO AND HISAO TOMINAGA Let R be a ring with prime radical P. The main theorems of this paper are (1) The following conditions are equivalent: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x 3 = x. (2) If R is a Pi-ring in which every element is the sum of two idempotents, then RjP satisfies the identity x -z. (3) Let it be a semi-perfect ring in which every element is the sum of two idempotents. If is quasi-projective, then R is a finite direct sum of copies of GF[2) and/or GF(3).

Commutative feebly nil-clean group rings

“…Let us recall that a ring is said to be tripotent if each its element x satisfies the equation x 3 = x. These rings are necessarily commutative being also a subdirect product of a family of single or isomorphic copies of the fields Z 2 and Z 3 (see, e.g., [6]). Likewise, as 6 = 0 here, any tripotent ring R can be decomposed as the direct product of two rings R 1 × R 2 , where R 1 is boolean and R 2 is tripotent of characteristic 3.…”

Tripotents: a class of strongly clean elements in rings

“…If all elements are tripotents (rings which satisfy the identity x 3 = x were investigated by Hirano and Tominaga -see [7]), the ring may not be Boolean (e.g. F 3 ).…”

Rings in which every unit is a sum of a nilpotent and an idempotent

“…We also deal with a special case where every unit of the ring is a sum of two commuting idempotents. These conditions can be compared with the so-called strongly 2nil-clean rings introduced by Chen and Sheibani [4], and the rings for which every element is a sum of two commuting idempotents, studied by Hirano and Tominaga in [11].We write M n (R), T n (R) and R[t] for the n × n matrix ring, the n × n upper triangular matrix ring, and the polynomial ring over R, respectively. For an endomorphism σ of a ring R, let R[t; σ] denote the ring of left skew power series over R. Thus, elements of R[t; σ] are polynomials in t with coefficients in R written on the left, subject to the relation tr = σ(r)t for all r ∈ R. The group ring of a group G over a ring R is denoted by RG.…”

“…This is by Theorem 3.6 and [4, Theorem 4.5].3.2. Units being sums of two idempotents.Hirano and Tominaga[11] have characterized the rings in which every element is the sum of two commuting idempotents.Definition 3.9. A ring R is called a UII-ring if every unit of R is a sum of two idempotents, and R is called a strong UII-ring if every unit of R is a sum of two commuting idempotents.Lemma 3.10.…”