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# nov09 - Math 5125 Wednesday November 9 November 9 Ungraded...

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Math 5125 Wednesday, November 9 November 9, Ungraded Homework Exercise 10.3.12 on page 356 Let R be a commutative ring with a 1 and let A , B and M be R -modules. Prove the following isomorphisms of R -modules. (a) Hom R ( A B , M ) = Hom R ( A , M ) Hom R ( B , M ) . (b) Hom R ( M , A B ) = Hom R ( M , A ) Hom R ( M , B ) . The proofs of (a) and (b) are similar; we will only prove (a). If γ Hom R ( A B , M ) , define α ( γ ) Hom R ( A , M ) by α ( γ )( a ) = γ ( a , 0 ) and β ( γ ) Hom R ( B , M ) by β ( γ )( b ) = γ ( 0 , b ) . If δ Hom R ( A B , M ) , then α ( γ + δ )( a ) = ( γ + δ )( a , 0 ) = γ ( a , 0 )+ δ ( a , 0 ) = α ( γ )( a )+ α ( δ )( a ) = ( α ( γ )+ α ( δ ))( a ) for all a A and we see that α ( γ + δ ) = α ( γ )+ α ( δ ) . Also if r R , then ( α ( r γ ))( a ) = ( r γ )( a , 0 ) = γ ( ra , 0 ) = ( α ( γ ))( ra ) = ( r ( α ( γ )))( a ) for all a A , hence α ( r γ ) = r ( α ( γ )) and we deduce that α is an R -module homomorphism. Similarly β is an R -module homomorphism. Finally we de- fine an R -module homomorphism θ : Hom R ( A B , M ) Hom R ( A , M ) Hom R ( B , M ) by θ ( γ ) = ( α ( γ ) , β ( γ )) . Now we want to define the inverse map to θ . If f Hom R ( A , M ) , define f Hom R ( A B , M ) by f ( a , b ) = f
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