Integral domain

Condition for a quotient commutative ring to be an integral domain

Let be a commutative ring and be a proper, nontrivial (two-sided) ideal. Then the quotient ring is an integral domain iff is a prime ideal. #m/thm/ring

Proof

Assume is an integral domain and let . Then , so either or .

For the converse, assume is prime. Since is automatically a commutative ring, it only remains to show that has no zero-divisors. To this end, assume . Then and hence or , whence or .

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