# Thread:

1. ## Affine varieties

(1) 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. (ascending chain condition).

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

No idea! Any help would be appreciated, thanks!

2. Originally Posted by shadow_2145
(1) 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. (ascending chain condition).

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

No idea! Any help would be appreciated, thanks!
you'll find complete and easy to understand solution to your problem in here.
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