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 $𝐽 = 𝑅$ . #m/def/ring

𝐼 ⊴ 𝐽 ⊴ 𝑅 ⟺ [𝐼 = 𝐽] \lor [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
$𝑅 / 𝐼$ for commutative $𝑅$ is a field iff $𝐼$ is maximal
Assuming choice, commutative ring has a maximal ideal

#state/tidy | #lang/en | #SemBr