TITLE: Can a Prime Ideal be equal to the parent Ring?
QUESTION [5 upvotes]: If $I$ is a prime ideal of $R$, can $I=R$?

REPLY [6 votes]: No. By definition, a prime ideal is a proper ideal, meaning that it is not the entire ring. This is mostly because there are theorems about prime ideals that don't work for the entire ring, and so it's easier to assert that prime ideals are not the whole ring than to put a "Let $I$ be a prime ideal of $R$ which is not equal to the whole ring" in many theorems and proofs.