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Quotient Rings and Homomorphisms

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Let R and S be rings. Show that [tex]\pi[/tex]:RxS->R given by [tex]\pi[/tex](r,s)=r is a surjective homomorphism whose kernel is isomorphic to S.


    2. Relevant equations



    3. The attempt at a solution
    To show that [tex]\pi[/tex] is a homomorphism map, I need to show that it's closed under addition and multiplication. And to show that it's surjective, I need to show for every x in RxS, f(x)=y for y in R. What does it mean when the kernel is isomorphic to S?
     
  2. jcsd
  3. Mar 8, 2010 #2
    You need to review your definitions of ring homomorphism and surjective function. Neither one that you gave is accurate.
     
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