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A fractional ideal of the integers with positive and negative prime exponents

Example

In $Z$ , the fractional ideal

I : = \frac{12}{35} Z

has factorization

I = (2)^{2} (3) (5)^{- 1} (7)^{- 1} .

Facts & Assumptions

Given: The domain $Z$ and the fractional ideal $I = (12 / 35) Z$ .

[F1]

A fractional ideal is a bounded nonzero submodule of the fraction field (Fractional ideals).

[L1]

Nonzero fractional ideals of a Dedekind domain factor uniquely into prime powers (Unique factorization of nonzero fractional ideals into prime powers).

Verification

technique · direct

1.1

The ideal $I$ is fractional because $35 I = 12 Z \subseteq Z$ , so [F1] applies.

F1given

2.1

In the PID $Z$ , the principal ideal generated by $12 / 35$ records the usual prime factorization of the numerator and denominator. Hence the exponents are $2$ at $(2)$ , $1$ at $(3)$ , $- 1$ at $(5)$ , and $- 1$ at $(7)$ , with all others zero. By [L1] this is exactly the prime-ideal factorization of $I$ .

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. P. May, Notes on Dedekind Rings (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, §20 (standard reference, not scraped)