The group ring of a ring.

It would be better to write elements of $R[R]$ in the form $$x=\sum_{s\in R}r_se^s.$$ Since $e^se^t=e^{s+t}$, multiplication in $R[R]$ captures the group structure of $(R,+)$. It also avoids the confusion you are having.

It's important to remember that the linear combinations are formal in that the way we write it distinguishes coefficients from generators: the $r_i$'s are coefficients, and the $s_i$'s are basis elements.

Their juxtaposition does not denote multiplication in $R$, but rather that $r_i$ is the coefficient at the base element $s_i$.

One can form a group ring over the additive group $(R,+)$ or a monoid ring over the monoid $(R,\cdot)$, so the notation above is a little ambiguous. It would perhaps be beneficial to just forget that $R$ is a ring and talk about its underlying abelian group $A$ (or use $M$ if you're doing the monoid instead.)