Principal Ideal Domain fulfills Ascending Chain Condition
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Theorem
Let $R$ be a principal ideal domain.
Then $R$ fulfils the ascending chain condition.
Proof
Let $I_1 \subseteq I_2 \subseteq I_3 \subseteq \dotsb$ be an ascending chain of ideals.
Build $\ds I = \bigcup_{i \mathop = 1}^\infty I_i$.
$I$ is an ideal.
Since $R$ is a principal ideal domain, $I = \ideal a$ for some $a \in R$.
Now, since $a \in I$, there is some $n$ such that $a \in I_n$.
Thus:
- $\ideal a \subseteq I_n$
By definition $I_n \subset I = \ideal a$, and so $I_n = I$.
Thus:
- $\forall m \ge n: I_m = I$
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$\blacksquare$
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