All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper by Ohm and Pendleton
My question is : If $R$ satisfies a.c.c. on radical ideals, then is it true that so does $R[[X]]$ ? If this is not true in general, is there any additional hypothesis on $R$ that would force this ?