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.

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A (-1, 1) ring \(R\) contains a maximal ideal \(I_{3}\) in the nucleus \(N\). The set of elements \(n\) in the nucleus which annihilates the associators in (-1, 1) ring \(R\), \(n(x, y, z) = 0\) and \((x, y, z)n = 0\) for all \(x, y, z \in R\) form the ideal \(I_{3}\) of \(R\). Let \(I\) be a right ideal of a 2-torsion free (-1, 1) ring \(R\) with commutators in the middle nucleus. If \(I\) is maximal and nil, then \(I\) is a two sided ideal. Also if \(I\) is minimal then it is either a two-sided ideal, or the ideal it generates is contained in the middle nucleus of \(R\) and the radical of \(R\) is contained in \(P\) for any primitive ideal \(p\) of \(R\).