Product ideal

Let $𝐼, 𝐽 ⊴ 𝑅$ be ideals (or fractional ideals). Their product ideal $𝐼 𝐽 = ⟨ 𝐼 𝐽 ⟩$ is the ideal given by the additive closure of $𝐼 𝐽$ . #m/def/ring

Properties

In what follows, $𝐼$ with or without a subscript will be some nonzero proper integral ideal, $𝔭$ with or without a subscript will be some nonzero prime ideal.

$𝐼 2 ∣ 𝐼 1$ implies $𝐼 1 \subseteq 𝐼 2$ ;
$I 1 \dots 𝐼 𝑛 \subseteq 𝔭$ implies $𝐼 𝑘 \subseteq 𝔭$ for some $𝑘 \in ℕ 𝑛$ .

Proof

Suppose $(𝐼 2 𝐼 3) = 𝐼 1$ . Since $(𝐼 2 𝐼 3) \subseteq 𝐼 2$ , it follows $𝐼 1 \subseteq 𝐼 2$ , proving ^D1.

Suppose towards contradiction there exists some $𝛼 𝑘 \in 𝐼 𝑘 ∖ 𝔭$ for all $𝑘 \in ℕ 𝑛$ . Then $\prod 𝑛 𝑘 = 1 𝛼 𝑘 \in (𝐼 1 \dots 𝐼 𝑛) \subseteq 𝔭$ , so since $𝔭$ is prime, some $𝛼 𝑘 \in 𝔭$ , a contradiction. Thus $(𝐼 1 \dots 𝐼 𝑛) \subseteq 𝔭$ implies $𝐼 𝑘 \subseteq 𝔭$ for some $𝑘 \in ℕ 𝑛$ , proving ^D2.

See also ^P1. These become iffs for a Containment-division ring, including a Dedekind domain.

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