Suppose A is a Noetherian ring, phi:A->A any surjective ring homomorphism. Show that phi is also injective. Also, if all the prime ideals of a ring A are finitely generated then is A noetherian? I'm pretty sure it is. I figure I can take all of the ideals that are not finitely generated and find a maximal prime ideal that contains these ideals. I've just started reading a book on commutative algebra with the hopes of moving on to algebraic geometry.