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Show that an integral domain $A$ is a principal ideal domain if every ideal $I$ of $A$ is principal, that is, of the form $I=(a)$. Show directly that the ideals in a PID satisfy the a.c.c.

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    I can show that, but since I don't know what definitions you use, I would say the first part of the statement is trivial (a PID is an integral domain where every ideal is prinipical, according to the definition I know). For the second part, just follow the standard proof.2012-05-02

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