Let u in A be a unit of the ring A. Let f A-A be defined by the fo.pdf

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Let u in A be a unit of the ring A. Let f: A->A be defined by the formula f(a)=(u^(-1))au. Show that f is an automorphism of A. Solution (a) You have to show the map i_u(r) preserves both addition and multiplication. Addition: i_u(r + s) = u(r + s)u^-1 = uru^-1 + usu^-1 = i_u(r) + i_u(s). Multiplication: I_u(rs) = ursu^-1 = uru^-1usu^-1 = i_u(r)i_u(s). ---------------------------------------... (b) If f is in Aut(R) we want to show f ? i_u ? f^-1 = i_u\' for some unit u\' - this will prove Inn(R) is normal in Aut(R). Computing f ? i_u ? f^-1(1) = f(u), since f^-1(1) = 1. Note that since u is a unit and f is an automorphism then f(u) is also a unit. Write u\' = f(u), we will show that f ? i_u ? f^-1 = i_u\'. Write f(s) = r, then: f ? i_u ? f^-1(r) = f(i_u(f^-1(r))) = f(i_u(s)) = f(usu^-1) = f(u)f(s)f(u^-1) = f(u)rf(u)^-1 = i_u\'(r). Which proves Inn(R) is normal in Aut R..

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