fractional ideal of a number field

数域的分式理想

Number Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: number-field fractional ideal

Translation evidence: Literature usage

Meaning in context

For a number field $K$, a nonzero finitely generated $\mathcal O_K$-submodule of $K$; equivalently it becomes an integral ideal after multiplication by a nonzero denominator in $\mathcal O_K$.

数域 $K$ 的一个非零有限生成 $\mathcal O_K$-子模；等价地，乘以某个非零的整数环元素后成为整理想。

Example

$\tfrac12\mathbb Z[i]$ is a fractional ideal of $\mathbb Q(i)$.

Usage and distinctions

A fractional ideal need not be contained in the integer ring; an integral ideal is.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义Section 1.3.1, Definition 1.33, printed page 14, specialized to O_K · Checked 2026-10-14

Translation sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义§1.3.1 定义1.33 分式理想，printed page 14 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.