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