Exercise 1.6. Let 1,!) : R > S be a ring homomorphism. Assume that R and S are local rings with maximal ideals M and N respectively. Show that the following are equivalent. (1) w is a local ring homomorphism; (ago/M N (3)1!) (N=) M; (4) for any x E R if 1b($) 1s invertible in S, then m is invertible in R