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Commutative rings

  1. May 6, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that a finite commutative ring with no zero divisors is an integral domain (i.e. contains a unity element)


    2. Relevant equations
    If a,b are elements in a ring R, then ab=0 if and only if either a and b are 0.


    3. The attempt at a solution
    I've been trying to use the cancellation laws.
     
  2. jcsd
  3. May 6, 2009 #2

    Dick

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    Let N be the set of all nonzero elements of the ring R. Pick a nonzero element c. Can you show cN=N? Remember N is a finite set.
     
  4. May 6, 2009 #3

    Dick

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    What do you mean? R is a ring. A 'ring' is closed under its multiplication operation.
     
  5. May 6, 2009 #4

    Dick

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  6. May 6, 2009 #5

    Dick

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    The clue is in post 2. Can you show multiplication by any nonzero element c maps the set of nonzero elements of the ring to itself in a one-to-one manner.
     
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