Commutative rings with identity

  • Thread starter Marinela
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I have a trouble proving that a finate (nonzero) commutative ring with no zero divisors must have an identity with respect to multiplication. Could anybody please give me some hints?
I do know all the definitions (of ring, commutative ring, zero divisors, identity) but have no idea how to go from there.
Thanks!
 

morphism

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Take some nonzero element a in your ring, and look at its various powers: a, a^2, a^3, .... Now use the fact that the ring has finitely many elements.

Edit:
Maybe this is not the easiest approach. It might be cleaner if you define a map x->ax on your ring. This is an injection, and hence a surjection (by finiteness). In particular, a=ax', for some x in the ring. Claim: x' is the multiplicative identity.
 
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