on the a.c.c. on radical ideals and the finite generation of radical ideals property of commutative rings with unity

Question

For commutative rings, consider the following two conditions :

1) ascending chain condition on radical ideals.

2) every radical ideal is finitely generated.

Does any of the conditions (1) or (2) imply the other for commutative rings ?

Answer 1

(For commutative rings) $2\implies 1$ because having all radical ideals finitely generated implies all prime ideals are finitely generated, and the well-known theorem called Cohen's theorem says this is an equivalent condition to being Noetherian, so it satisfies the ACC on all ideals.

On the other hand, the converse is false because it is easy to produce non Noetherian rings with a unique prime ideal: such a ring clearly satisfies 1 but not 2. For example, take $V=\oplus_{i=1}^\infty F$ for a field $F$, and form the trivial extension $R=F(+)V$ with pointwise addition and multiplication given by $(a,v)(b,w)=(ab, aw+bv)$. Every subspace of $\{0\}(+)V$ is an ideal, so $R$ fails the ACC on ideals. This same ideal is the unique prime ideal of $R$, since it is both maximal and nilpotent.