Suppose $I$ and $J$ are finitely generated ideals in comm ring $R$ then $I+J$ is finitely generated.
Question is converse for this.
Suppose $I+J$ is finitely generated then $I$ and $J$ are finitely generated?
Suppose $I+J$ is finitely generated and $I$ is finitely generated then $J$ is finitely generated?
Suppose $I+J$ is finitely generated and $I$ is nilpotent finitely generated then $J$ is finitely generated?
I tried proving, failed. Tried to give some counterexamples, failed.
I am sure at least third point is true.
Suggest some hints. Any remarks regarding this if not hints are most welcome.