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The aim of this work is to study a decomposition theorem for ringssatisfying  either of the  properties xy=xpf(xyx)xq or xy=xpf(yxy)xq, where p=p(x,y),q=q(x,y) are nonnegativeintegers and f(t)∈tℤ[t] vary with the pair ofelements x,y, and further investigate the commutativity of suchrings. Other related results are obtained for near-rings

On structure of certain periodic rings and near-rings