apr13 - { ( 2 a , 2 b ) | a , b Z } (c) { ( 2 a , ) | a Z }...

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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)
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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 ....
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