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.
1
Expert's answer
2012-10-31T08:48:09-0400
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.
Numbers and figures are an essential part of our world, necessary for almost everything we do every day. As important…
APPROVED BY CLIENTS
"assignmentexpert.com" is professional group of people in Math subjects! They did assignments in very high level of mathematical modelling in the best quality. Thanks a lot
Comments
Leave a comment