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June 2nd, 2017, 06:20 PM   #1
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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 ?
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June 2nd, 2017, 09:01 PM   #2
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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.
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June 2nd, 2017, 09:22 PM   #3
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Quote:
Originally Posted by Maschke View Post
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
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June 2nd, 2017, 10:02 PM   #4
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Quote:
Originally Posted by Lalitha183 View Post
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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