Ideal

Product ideal

Let be ideals (or fractional ideals). Their product ideal is the ideal given by the additive closure of . ring

Properties

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

1. implies ;
2. implies for some .

Proof

Suppose . Since , it follows , proving ^D1.

Suppose towards contradiction there exists some for all . Then , so since is prime, some , a contradiction. Thus implies for some , proving ^D2.

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

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