Can R[[X]] be artin ring？

Question

Let R be ring. Is R[[X]] can be artin ring？

In my image, both artin ring and formal power series ring are similar to field(but both are not field).

Answer 1

Assuming that $R$ is a nonzero ring, the power series ring $R[[X]]$ cannot be an Artinian ring.

This is simply because the descending chain of ideals $(X) \supset (X^2) \supset (X^3) \supset \cdots$ is not stable.

In geometric intuition, the spectral of $R[[X]]$ is one dimensional higher than that of $R$, while the dimension of an Artinian ring is zero.