Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 101 / 4 / d / Solution

No. For example, take $R=k$, any field. It is an Artinian ring, since its only ideals are zero and itself. In the formal power series ring $k[[x]]$, however,is a strictly descending chain of ideals. The inclusion is strict because $x^n$ has a nonzero coefficient of $x^n$, whereas every element of $(x^{n+1})$ has that coefficient zero. Thus $k[[x]]$ fails the descending chain condition and is not Artinian.

The same argument works for any nonzero coefficient ring $R$: the coefficient $1$ in $x^n$ cannot be produced by a multiple of $x^{n+1}$. Therefore $R[[x]]$ is never Artinian when $R\ne 0$. The zero ring is the harmless exception.