TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03

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Every Euclidean domain is a principal ideal domain

Statement

Every Euclidean domain is a principal ideal domain.

Facts & Assumptions

Given: A Euclidean domain $R$ with Euclidean function $\delta$ , and an ideal $I\mathrel{\trianglelefteq}R$ .

[L1]

Euclidean division gives $a=bq+r$ with $r=0$ or $\delta(r)<\delta(b)$ whenever $b\ne0$ (Euclidean domain and Euclidean function).

[L2]

An ideal is an additive subgroup closed under multiplication by arbitrary ring elements (Left, right and two-sided ideals).

[L3]

The principal ideal $(d)$ is the ideal generated by $d$ (The ideal generated by a subset and principal ideals).

[L4]

Every nonempty subset of $\mathbb N$ has a least element (The well-ordering principle).

[L5]

A PID is an integral domain whose every ideal is principal (Principal ideal domain).

Proof

technique · direct

1.1

If $I=\{0\}$ , then $I=(0)$ and is principal.

1.2

Suppose $I\ne\{0\}$ . The set $\{\delta(x):x\in I\setminus\{0\}\}$ is nonempty, so choose $d\in I\setminus\{0\}$ whose $\delta$ -value is least.

L4givenchoose

2.1

For $a\in I$ , divide by $d$ : $a=dq+r$ with $r=0$ or $\delta(r)<\delta(d)$ . Since $r=a-dq\in I$ , minimality in step 1.2 excludes a nonzero $r$ ; hence $r=0$ .

step 1.2L1L2given

3.1

Step 2.1 gives $a=dq\in(d)$ for every $a\in I$ , so $I\subseteq(d)$ . Conversely $d\in I$ and ideal closure give $dq\in I$ for every $q\in R$ , so $(d)\subseteq I$ . Thus $I=(d)$ .

step 2.1L2L3given

4.1

Every ideal is principal by step 1.1 or step 3.1; therefore $R$ is a PID.

step 1.1step 3.1L5∎

Depends on

Used by

For every field F, F[x] is a principal ideal domain Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 38 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)