$R$ is a commutative ring with $I \lhd R, a \in R$
If $I + Ra$ is finitely generated, I want to show that this implies that $\exists$ finitely generated $J \subseteq I$ finitely generated with $J + Ra = I + Ra$.
I have been attempting to prove this for a while now, as I have used it to prove that if every prime ideal of $R$ is finitely generated then $R$ is necessarily Noetherian; but embarrassingly I can't seem to get this first statement.
Any help appreciated!