Ideal

Prime ideal

A (two-sided) proper ideal 𝔭 ◃𝑅 is called a prime ideal iff for any 𝑎,𝑏 ∈𝑅, 𝑎𝑏 ∈𝔭 implies 𝑎 ∈𝔭 or 𝑏 ∈𝔭¹, ring i.e.

𝔭∋𝑎𝑏⟺[𝔭∋𝑎]∨[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∣𝑎𝑏⟺[𝓅∣𝑎]∨[𝓅∣𝑏]

i.e. 𝓅 is prime.

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

Properties

• Condition for a quotient commutative ring to be an integral domain (often this is used to prove primality)
• A maximal ideal in a commutative ring is prime
• ^D2

See also

• Krull dimension
• Relatively prime ideals
• Prime order of an ideal

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Footnotes

1. 2017. Contemporary abstract algebra, §14, p. 253 ↩