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**shadow_2145** (1) An integral domain $\displaystyle A$ is a *principal ideal domain *if every ideal $\displaystyle I$ of $\displaystyle A$ is principal, that is of the form $\displaystyle I = (a)$; show directly that the ideals in a PID satisfy the a.c.c. (ascending chain condition).

(2) Show that an integral domain $\displaystyle A$ is a UFD if and only if every ascending chain of principal ideals terminates, and every irreducible element of $\displaystyle A$ is prime

No idea! Any help would be appreciated, thanks!