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 June 2nd, 2017, 06:20 PM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 Reals Let $R$ be a finite commutative ring with no zero divisors then A) $R$ is a field B) $R$ has unity C) Characteristic of $R$ is prime number D) None of the above Option A & B are true and I got confused with Option C. I know that every field has characteristic of either $0$ or prime and the finite field has characteristic of prime. Is $R$ a finite field ?
 June 2nd, 2017, 09:01 PM #2 Senior Member   Joined: Aug 2012 Posts: 1,887 Thanks: 524 For A, a finite commutative ring must have unity. https://math.stackexchange.com/quest...y-have-a-unity Then if it has no zero divisors it's an integral domain. It's well known that a finite integral domain is a field. That latter's an easy proof.
June 2nd, 2017, 09:22 PM   #3
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 Originally Posted by Maschke For A, a finite commutative ring must have unity. https://math.stackexchange.com/quest...y-have-a-unity Then if it has no zero divisors it's an integral domain. It's well known that a finite integral domain is a field. That latter's an easy proof.
Since $R$ is a finite commutative Ring with no zero divisors, it has unity in it. Which implies $R$ is an Integral domain. As you said a finite integral domain is a field then is this field finite or Infinite ?
Depending on it I can choose option C.

Thank you

June 2nd, 2017, 10:02 PM   #4
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 Originally Posted by Lalitha183 Since $R$ is a finite commutative Ring with no zero divisors, it has unity in it. Which implies $R$ is an Integral domain. As you said a finite integral domain is a field then is this field finite or Infinite ? Depending on it I can choose option C. Thank you
Doesn't yousay it's finite? An infinite integral domain need not be a field, just take the integers.

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