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An ideal is prime exactly when the corresponding quotient ring has no zero divisors.
Prime ideal iff quotient is integral domain: Let be a commutative ring with , and let be an ideal. Then is a prime ideal if and only if the quotient ring is an integral domain.
This is the fundamental translation between multiplicative primeness and “domain-like” behavior after quotienting.
An ideal in a ring is an additive subgroup such that it is stable under multiplication by elements of on one side:
Let be a ring and let be a two-sided ideal of . The quotient ring is the quotient set of by the equivalence relation iff , with addition and multiplication on cosets defined by
The two-sided condition ensures multiplication is well-defined.
The canonical projection is a surjective ring homomorphism with kernel , and satisfies the universal property of quotients.
An integral domain is a commutative ring with such that for all , implies or (equivalently, has no zero divisors).