This is the problem I wish to discuss: An ideal that is maximal among non-finitely generated ideals is prime.
The strategy seems to be to assume that there exists some maximally non-f.g. ideal that is not prime, and then reach a contradiction.
At no point in the given solutions do the authors consider the possibility that the maximally non-f.g. ideal in question could be equal to the whole ring. The accepted answer starts "suppose $U$ is not prime, then $\exists x, y \in R: xy \in U, x \notin U$, $ y\notin U$". This ignores the other possibility that $U = R$.
Can anyone shed some light on this issue for me? I am having trouble following several similar proofs. Thanks.
