Commutative rings

  • #1

Homework Statement


Show that a finite commutative ring with no zero divisors is an integral domain (i.e. contains a unity element)


Homework Equations


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


The Attempt at a Solution


I've been trying to use the cancellation laws.
 

Answers and Replies

  • #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.
 
  • #3
Dick
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thanks, but we can't assume R is closed under multiplication.
What do you mean? R is a ring. A 'ring' is closed under its multiplication operation.
 
  • #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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