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Problem 64 (Ascending Chains of Ideals In a PID Are Finite)

Suppose that $R$ is a ring, and that $A_1\subseteq A_2\subseteq A_3\subseteq \cdots$ is nested sequence of ideals of $R$.

  1. Prove that the union $\ds \bigcup_{n=1}^{\infty} A_n$ is an ideal of $R$.
  2. If $R$ is a principle ideal domain, prove that there exists an integer $k$ so that $A_k=A_n$ for every $n\geq k$. In other words, prove that every ascending chain of ideals in a principle ideal domain contain finitely many different ideals.
  3. Challenge: Give an example of a ring $R$ and an infinite ascending chain of ideals so that no two ideals are equal.

You can present in class even if you don't get the third part.

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