# Thread: Affine varieties

1. ## Affine varieties

(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!

2. Originally Posted by 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!
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