Maximal ideal

A (two-sided) proper ideal is called a maximal ideal iff whenever is an ideal such that , then either or . #m/def/ring

Maximal ideal generalizes Irreducible element in the same way that Prime ideal generalized Prime element.

Properties

• A maximal ideal in a commutative ring is prime
• for commutative is a field iff is maximal
• Assuming choice, commutative ring has a maximal ideal

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