# apr13 - 2 a 2 b | a b ∈ Z(c 2 a | a ∈ Z(d a-a | a ∈ Z...

This preview shows page 1. Sign up to view the full content.

Math 4124 Wednesday, April 13 April 13, Ungraded Homework Exercise 7.3.6 on page 248. Decide which of the following are ring homomorphisms from M 2 ( Z ) to Z . (a) ± a b c d ² 7→ a (b) ± a b c d ² 7→ a + d (c) ± a b c d ² 7→ ad - bc Let us call the relevant homomorphism θ . (a) No, because multiplication is not respected If x = ± 0 1 0 0 ² and y = ± 0 0 1 0 ² , then θ ( x ) θ ( y ) = 0 * 0 = 0, but θ ( xy ) = 1. (b) No, because multiplication is not respected. In fact taking x , y as above, we get θ ( x ) θ ( y )= 0 * 0 = 0, but θ ( xy ) = 1. (c) No, because addition is not respected. In fact taking x , y as above, we get θ ( x )+ θ ( y ) = 0 + 0 = 0, but θ ( x + y ) = - 1. Exercise 7.3.8 on page 248. Decide which of the following are ideals of the ring Z × Z . (a) { ( a , a ) | a Z } (b)
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: { ( 2 a , 2 b ) | a , b ∈ Z } (c) { ( 2 a , ) | a ∈ Z } (d) { ( a ,-a ) | a ∈ Z } Let the relevant set of elements be I . (a) No. For example ( 1 , 1 ) ∈ I , but ( 1 , 1 )( 1 , ) / ∈ I . (b) Yes. 0 ∈ I and I is closed under +. Finally if ( x , y ) ∈ Z × Z , then ( 2 a , 2 b )( x , y ) = ( 2 ax , 2 by ) ∈ I . (c) Yes. 0 ∈ I and I is closed under +. Finally if ( x , y ) ∈ Z × Z , then ( 2 a , )( x , y )=( 2 ax , ) ∈ I . (d) No. For example ( 1 ,-1 ) ∈ I , but ( 1 ,-1 )( 1 ,-1 ) = ( 1 , 1 ) / ∈ I ....
View Full Document

## This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.

Ask a homework question - tutors are online