Let $R$ is a local ring, from the depth lemma, we can get $\operatorname{depth}(R\oplus\dotsb\oplus R)=\operatorname{depth}(R)$, here the direct sum is finite, how about the infinite case?
By the intuition, at least for the countable $R$, the result is also $\operatorname{depth}(R)$, but we have depth $R[x]=\operatorname{depth}(R)+1$, any difference between the $R[x]$ and the countable direct sum of $R$?
Let $R$ be a local ring. If $M$ is faithfully flat over $R$, then $\text{depth}_R(M) = \text{depth}(R)$, since for any $R$-ideal $I$ and any $x \in R$,
$$0 \to R/I \xrightarrow{x} R/I \to R/(I, x) \to 0$$
is exact iff
$$0 \to M/IM \xrightarrow{x} M/IM \to M/(I, x)M \to 0$$
is exact, so a sequence of elements $x_1, \ldots, x_n$ is regular on $R$ iff it is regular on $M$ (note that the first map gives weak regularity, and the second map improves this to full regularity).