Prime ideal

A (two-sided) proper ideal $𝔭 ◃ 𝑅$ is called a prime ideal iff for any $𝑎, 𝑏 \in 𝑅$ , $𝑎 𝑏 \in 𝔭$ implies $𝑎 \in 𝔭$ or $𝑏 \in 𝔭$ ¹, #m/def/ring i.e.

𝔭 ∋ 𝑎 𝑏 ⟺ [𝔭 ∋ 𝑎] \lor [𝔭 ∋ 𝑏]

Historical note

Considering the original notion of an Ideal number, an ideal $𝔭$ is the set of multiples of an ideal number $𝓅$ . Therefore the above is equivalent to

𝓅 ∣ 𝑎 𝑏 ⟺ [𝓅 ∣ 𝑎] \lor [𝓅 ∣ 𝑏]

i.e. $𝓅$ is prime.

Note an ideal $𝐼 ⊴ ℤ$ is prime iff $𝐼 = 𝑝 ℤ$ where $𝑝$ is prime or zero. The set of all prime ideals of a commutative ring is called its spectrum.

Properties

$𝑅 / 𝐼$ for commutative $𝑅$ is an integral domain iff $𝐼$ is prime (often this is used to prove primality)
A maximal ideal in a commutative ring is prime
^D2

Prime ideal

Properties

See also

Footnotes