This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 Rmodule 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 Rbilinear. (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 Zbilinear 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 =...
View
Full
Document
This note was uploaded on 01/02/2012 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.
 Fall '07
 PALinnell
 Math, Algebra

Click to edit the document details