Hisao Tominaga

Rings in which every element is the sum of two idempotents

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 R/P satisfies the identity x 3 = x . (3) Let R be a semi-perfect ring in which every element is the sum of two idempotents. If R R R is quasi-projective, then R is a finite direct sum of copies of GF (2) and/or GF (3).

A commutativity theorem for rings

We prove the following theorem: Let R be a ring, l a positive integer, and n a non-negative integer. If for each x , y ∈ R , either xy = yx or xy = x n f(y) x 1 for some f( X ) ∈ X 2 Z[ X ], then R is commutative.

Commutativity theorems for rings with polynomial constraints on certain subsets

We prove several commutativity theorems for unital rings with polynomial constraints on certain subsets, which improve and generalise the recent results of Grosen, and Ashraf and Quadri.

A commutativity theorem for semi-primitive rings

In this brief note, we prove the following: Let R be a semi-primitive ring. Suppose that for each pair x , y e R there exist positive integers m = m ( x , y ) and n = n ( x , y ) such that either [ x m ,( xy ) n − ( yx ) n ] = 0 or [ x m ,( xy ) n + ( yx ) n ] = 0. Then R is commutative.