Equational definability of addition in certain noncommutative rings (Q799783)

It is shown that if R is a ring with unity 1, not necessarily commutative which satisfies the identity \(x^ n=x^{n+1}f(x)\), where n is a fixed positive integer and f(x) is a fixed polynomial with integer coefficients, then the addition \(''+''\) of R is equationally definable in terms of multiplication ''\(\times ''\) and the successor operation ''{\^{\ }}'' (x{\^{\ }}\(=x+1)\).