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.
Integral domain and ascending chain condition proof
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commutative-algebra
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0I 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