Let a be an element in a ring such that ma = 0 = a^(2^r) , where m ≥ 1 and r ≥ 0 are given integers. Show that (1 + a)^(m^r) = 1.
The proof is by induction on r.The case r = 0 being clear, we assume r > 0. Since ma =0, the binomial theorem gives (1 + a)m= 1+a2bwhere b is a polynomial in a with integer coefficients. Sincem(a2b) = 0 and (a2b)^2r−1= a^2r*b^2r−1 = 0, theinductive hypothesis (applied to the element a2b)implies that 1 = (1+a2b)^mr−1= [(1 + a)m]^mr−1 = (1+a)^mras desired.