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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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