If $R\subset S$, does it necessarily follow that $R[G]\subset S[G]$?

Question

So perhaps this is a trivial question, but suppose we have some rings (possibly noncommutative) such that $R\subset S$, and a group (possibly infinite) $G$, does it necessarily follow that $R[G]\subset S[G]$?

Answer 1

It will be contained clearly. I guess you are asking will it be a subring.

So, yes it will be, as let $\alpha =\sum a_gg\ , \beta=\sum b_gg \in RG$ then $\alpha-\beta=\sum(a_g-b_g)g\in RG$ as $(a_g-b_g)\in R$ and also it is closed under multiplication as $\alpha\beta=\sum_{g,h\in G} a_gb_hgh$ as again $R$ is closed under multiplication so $a_gb_h\in R$