June 2nd, 2017, 06:20 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 154 Thanks: 1  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: 1,414 Thanks: 342 
For A, a finite commutative ring must have unity. https://math.stackexchange.com/quest...yhaveaunity 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  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 154 Thanks: 1  Quote:
Depending on it I can choose option C. Thank you  
June 2nd, 2017, 10:02 PM  #4 
Senior Member Joined: Aug 2012 Posts: 1,414 Thanks: 342  Doesn't yousay it's finite? An infinite integral domain need not be a field, just take the integers.


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