Could anyone please explain to me why not vice versa in this paragraph, and how are the rings $R[x_1,\dots,x_m]$ and $R[[x]]$ clarify this example? which is a subring from the other?How can I prove that they are $R-modules$? :
If $S$ is a ring and $R$ is a subring, then $S$ is an $R$-module (but not vice versa!) with
$ra$ $(r\in R, a\in S$) being multiplication in $S$. In particular, the rings $R[x_1,\dots,x_m]$ and $R[[x]]$ are $R$-modules.