Maximal ideal

A (two-sided) proper ideal 𝐼 ◃𝑅 is called a maximal ideal iff whenever 𝐽 ⊴𝑅 is an ideal such that 𝐼 ⊴𝐽 ⊴𝑅, then either 𝐼 =𝐽 or 𝐽 =𝑅. ring

𝐼⊴𝐽⊴𝑅⟺[𝐼=𝐽]∨[𝐽=𝑅]

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
• Condition for a quotient commutative ring to be a field
• Every commutative ring has a maximal ideal

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