Let $R$ be a commutative ring (not necessarily with unity) having a non-finitely generated ideal. Then is it true that there is a non-terminating descending chain of non-finitely generated ideals in $R$ ? If this is not true in general, what happens if we also assume that the ring have unity ?
Note that descending chain of non-finitely generated ideals shows that the claim is true if we assume that $R$ has no non-zero zero divisor.
No.
At https://mathoverflow.net/q/304149 you will find a description of a unital, commutative, local ring with exactly one nonfinitely generated ideal.