113 HW 8_Math 113 - Mathematics 113: Homework # 8 Curtis...

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Unformatted text preview: Mathematics 113: Homework # 8 Curtis Tekell Dr. Poirier November 3, 2010 Exercise 1: Let R and S be rings. Are the following homomorphisms? (a) f : r 7 r (b) f : r 7 r r (c) f : r s 7 r (d) f : r s 7 rs (e) f : r s 7 r + s Proof. Let a,b,c R , and let x,y,z S . (a) Yes. f ( ab + c ) = ( ab + c ) 0 = ( a 0)( b 0) + ( c 0) = f ( a ) f ( b ) + f ( c ). (b) Yes. f ( ab + c ) = ( ab + c ) ( ab + c ) = ( a a )( b b ) + ( c c ) = f ( a ) f ( b ) + f ( c ). (c) Yes. f (( a x )( b y )+( c z )) = f (( ab + c ) ( xy + z )) = ab + c = f ( a x ) f ( b y )+ f ( c z ). (d) No. We assume that R 6 = 0, else this is a homomorphism, and we let r R be nonzero. Then we have f ( r 0 + 0 r ) = f ( r r ) = r 2 6 = 0 = r 0 + 0 r = f ( r 0) + f (0 r ) (e) No. We have the following: f (( a x )( b y )) = f ( ab xy ) = ab + xy 6 = ( a + x )( b + y ) = f ( a x ) f ( b y )....
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113 HW 8_Math 113 - Mathematics 113: Homework # 8 Curtis...

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