TITLE: If a finitely-generated ideal is prime, does it follow that the generators are prime?
QUESTION [7 upvotes]: If a finitely-generated ideal is prime, does it follow that the generators are prime? This is clearly true if it is generated by $1$ element, but is it true for any $n \ge 2$? If not, are there prime ideals generated by elements, none of which are prime?

REPLY [2 votes]: It is not clear what you mean by the generators. In $\mathbb Z$, $(4, 6)=(2)$ is prime, but neither $4$ nor $6$ are.