Math Help - Affine varieties

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!

(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