Prime ideal

Prime order of an ideal

Let 𝑅 be a commutative ring, 𝔭 be a prime ideal, and 𝔞 be an ideal. Then ord𝔭⁡(𝔞) is the largest 𝑚 ∈ℕ0 such that 𝔭𝑚 ∣𝔞, ring see product ideal.¹ For 𝛼 ∈𝑅, we also write ord𝔭⁡(𝛼) :=ord𝔭⁡(⟨𝛼⟩).

Properties

1. ord𝔭⁡(𝔞𝔟) ≥ord𝔭⁡(𝔞) +ord𝔭⁡(𝔟), which becomes an equality if 𝑅 admits UFI.

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Footnotes

1. 2022. Algebraic number theory course notes, p. 26 ↩