= Solution
The formal power series ring $P=k[[X]]$ is Noetherian, so the finite product $R=P\times P$ is Noetherian. Its maximal ideals are
$$
(X)\times P,\qquad P\times(X),
$$
so there are exactly two.
The ideal
$$
\mathfrak p=(X)\times P=((X,1))
$$
is principal and prime because $R/\mathfrak p\simeq k$. The prime chain
$$
(0)\times P\subsetneq(X)\times P
$$
shows that it has height one, and no longer chain exists because $\dim P=1$. Yet
$$
(1,0)(0,1)=0
$$
with both factors nonzero, so $R$ is not a domain. This shows why locality is essential in part i.
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