Identities of a group ring

Question

Let $R$ be a ring with identity $1_R\neq 0$, and let $G$ be a finite group with group operation written multiplicatively. Then we can form the group ring $$RG = \{a_1g_1+a_2g_2+\cdots+a_ng_n\ |\ a_i\in R\}.$$ In Dummit & Foote's Abstract Algebra, they state that if $1_G$ is the identity of $G$, then we write $r1_G$ as $r$, and $1_Rg$ as $g$. But then, shouldn't we have: $$1_G = 1_R1_G = 1_R?$$ If that is the case, is there any harm in defining only one identity $1_{RG}$ for $RG$?

Answer 1

There's no harm in this. It's just notation.