A non-zero ring R will be called right a -ring if every right ideal of it is automorphism-invariant, that is, for any automorphism f : E(I) ?? E(I) , where E(I) is an injective envelop of I , we have f(I) ? I . The work in this thesis is promoted by “right a -ring”. The main result is given in the following theorem: Theorem. A right a -ring is a direct sum of a square-full semi simple Artinian ring and a right square-free. Also, we showed a ring R is semi simple Artinian if and only if the matrix ring is a right a -ring for some n gt; 1 . We will consider some special 1 implies yx = 1 for all x,y ? R. If for a ring R every matrix ring is directly-finite then R is called a stably-finite ring. Theorem. Every right a -ring is stably-finite.
