Thread: Affine varieties

(1) An integral domain is a principal ideal domain if every ideal of is principal, that is of the form ; show directly that the ideals in a PID satisfy the a.c.c. (ascending chain condition).

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

No idea! Any help would be appreciated, thanks!

Originally Posted by shadow_2145

(1) An integral domain is a principal ideal domain if every ideal of is principal, that is of the form ; show directly that the ideals in a PID satisfy the a.c.c. (ascending chain condition).

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

No idea! Any help would be appreciated, thanks!

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