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Abstract.In this paper we show that if R is a semiprime assosymmetric ring with ascending chain condition for the right annihilators of the form r(x), x∈N(R), and also R contains no infinite direct sums of nonzero right ideals then the right quotient ring of R relative to S exists and it is semisimple and artinian.

Let R be a semiprime nonassociative ring satisfying (x, y, z)–(z, y, x) ∈ Nr then Nl = Nr where Nl and Nr are Lie ideals of R, the set {x ∈ Nr : (R, R, R)x = 0} = {x ∈ Nl : x(R, R, R) = 0} is an ideal of R, and it is contained in the nucleus. Further if [R, R]Nr ⊂ Nr and R is a prime ring with Nr ≠ 0 then R is either associative or commutative.

A semiprime ring R of characteristic ≠ 2 satisfying the identities x(yz) = y(xz) and (xy)z = y(zx) must be associative and commutative. If (y, x, z) is replaced by − (y, x, z) even then R is associative.