a. A commutative ring R is said to be noetherian if every increasing chain, I_{1}⊆ I_{2}⊆ ... ⊆ I_{n}⊆ ... , of ideals stabilizes.

If R is Noetherian show that any surjective ring endomorphism f:R --> R is an isomorphism.

b. IsZNoetherian? Why or why not?