Simple (-1,1) rings with an idempotent

A generalization of (−1,1) rings

“…Of course (3) also holds in every right alternative ring. All three identities are valid in ( -1, 1) rings, which have been investigated in detail by Maneri [7] and Hentzel [1,2].…”

On right alternative rings without proper right ideals

“…A number of people have worked on this problem and were able to prove the alternative identity whenever they assumed an additional hypothesis such as finite dimensionality [1,3], other identities [6,7], or internal conditions on the ring [4,5,9]. It seems natural to try to tackle the case where there exists an idempotent e Φ 1 in R such that (β, e, R) = 0.…”

Peirce decomposition in simple Lie-admissible power-associative rings

“…Lie rings do not have idempotent elements, and simple (7, d) rings with an idempotent eφl have been shown [2,3,4,5,6,8] to be associative. Thus if A has an idempotent element e Φ 1 then A belongs to a class which includes rings of the associative, commutative power-associative, and antiflexible types.…”