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 https://math.stackexchange.com/questions/2887225/descending-chain-of-non-finitely-generated-ideals shows that the claim is true if we assume that $R$ has no non-zero zero divisor.
It is certainly false if you do not require the ring to have a unity.
Take an abelian group $G$ that satisfies DCC on subgroups but not ACC; for example, the Prufer $p$-group, $C_{p^{\infty}}$ is not finitely generated, so it does not satisfy ACC. However, it does satisfy DCC, since any proper subgroup is finite.
Now make it into a ring by giving it zero multiplication, $a*b = 0$ for all $a,b\in G$. This is a ring, ideals are the same thing as subgroups, and hence this ring satisfies DCC on ideals (so it does not have an infinite descending chain of ideals, and in particular it does not have any infinite descending chain of non-finitely generated ideals; in fact, the only non-finitely generated ideal is the whole ring), but not ACC (and the whole ring is an example of a non-finitely generated ideal).