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An ideal is maximal exactly when the corresponding quotient ring is a field.
Maximal ideal iff quotient is field: Let be a commutative ring with , and let be an ideal. Then is a maximal ideal if and only if the quotient ring is a field.
This equivalence is the main bridge between ideal-theoretic maximality and “no nontrivial quotients” in commutative algebra.
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 commutative ring with . A maximal ideal is a proper ideal such that there is no ideal with .
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.
A field is a commutative ring with such that every nonzero element is a unit (equivalently, every has a multiplicative inverse).