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Unformatted text preview: Math 5125 Wednesday, December 7 Twelfth Homework Solutions 1. Suppose R is a commutative ring and let I , J be ideals of R . (a) Show that there is a surjective R-module homomorphism from I R J to IJ map- ping i j to i j . (b) Give an example to show that the map in (a) need not be injective. (a) This is true because ( i , j ) 7 i j : I R J IJ is R-bilinear. (b) Let R = Z / 4 Z and I = J = 2 Z / 4 Z . Then IJ = 0, so all we need to do is to show I Z / 4 Z J 6 = 0. However the map : I J Z / 2 Z defined by ( 2 + 4 Z , 2 + 4 Z ) = 1 + 2 Z and to be 0 on other elements is Z / 4 Z-bilinear and not the zero map. This proves that I Z / 4 Z J 6 = 0 and we have the required example. 2. Section 10.5, Exercise 2 on page 383. Suppose that A --- B --- C --- D y y y y A --- B --- C --- D is a commutative diagram of groups with exact rows. Prove that (a) if is surjective and , are injective, then is injective. (b) if is injective and , are surjective, then is surjective. (a) Let c ker . Then c = 0 and hence c = 0. Since is injective, we deduce that c = 0, consequently c = b for some b B . Then b = b = 0, so b =...
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