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Nathan Nguyen
2025-02-07 21:38:28 -06:00
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docs/math/abstract-algebra/18-integral-domains.md
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@@ -53,7 +53,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
**Lemma**. 18.14: Let $D$ be a PID. Let $I_1 \subseteq I_2 \subseteq \ldots$. Then, there exists some integer $N$ such that $I_n = I_N$ for all $n > N$. That is, any chain of ideals converges.
**Definition**. Any commutative ring that satisfies the above condition (the *ascending chain condition*), even if it's not a PID, is called a *Noetherien ring*.
**Definition**. Any commutative ring that satisfies the above condition (the *ascending chain condition*), even if it's not a PID, is called a *Noetherian ring*.
**Theorem**. 18.15: Every PID is a UFD. Note that the converse is not true.