nov09 - Math 5125 Wednesday, November 9 November 9,...

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Unformatted text preview: 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 Hom R ( A B , M ) , define ( ) Hom R ( A , M ) by ( )( a ) = ( a , ) and ( ) Hom R ( B , M ) by ( )( b ) = ( , b ) . If Hom R ( A B , M ) , then ( + )( a ) = ( + )( a , ) = ( a , )+ ( a , ) = ( )( a )+ ( )( a ) = ( ( )+ ( ))( a ) for all a A and we see that ( + ) = ( )+ ( ) . Also if r R , then ( ( r ))( a ) = ( r )( a , ) = ( ra , ) = ( ( ))( ra ) = ( r ( ( )))( a ) for all a A , hence ( r ) = r ( ( )) and we deduce that...
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