# On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

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 ?

• I have no time for a closer look at the moment, but the examples at the end of J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177, 299-304 might be of interest. – Fred Rohrer Feb 19 at 21:22
• In the aforementioned paper it is proved the following: If $V$ is a rank one non-discrete valuation ring, then $V$ has Noetherian spectrum, while $V[[X]]$ has an infinite chain of prime ideals, and therefore it does not have Noetherian spectrum. – user26857 Mar 4 at 19:04