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Theorem 1: If R be a Non-associative ring with unity 1 such that for (xy) = (xy)x all x,y in R, then R is commutative. Proof: Given identity is (xy) = (x y)x Replacing x by x+1 in the given condition, [(x+1)y] = [(x+1)y](x+1) (xy+y)=[(xy+y)(x+1)] (xy+y) (xy+y)=[(xy 2 )x+xy+yx+y] (xy)+(xy)y+y(xy)+y=(xy 2 )x+(xy)y+yx+y (by the condition and cancellation law) (xy)y+y(xy)= (xy)+yx Replacing y by y+ 1,[x(y+1)] (y+1)+(y+1)[x(y+1)] =x(y+2y+1)+( (y+2y+1)x We get xy=yx. ∀ x,y ∈ R(by the condition and cancellation law) Hence R is commutative ring.

Three Results for Non-Associative Semi Prime Ring with Unity.