If $R[x]=R \oplus \langle x \rangle \oplus \langle x^2 \rangle \oplus \cdots $ is a grading of $R[x]$, and $r \in R$, then where does $rx$ belong?

Question

A graded ring is a ring $R$ with a decomposition $R=\bigoplus_{i \ge 0} R_i$ of the abelian group $(R, +)$ into a direct sum of abelian groups $R_i$ such that $R_i \cdot R_j \subset R_{i+j}$.

Consider $R[x]=R \oplus \langle x \rangle \oplus \langle x^2 \rangle \oplus \cdots$. Since each $R_i$ is just an abelian group, we can only add elements. So, for instance, $r \notin \langle x \rangle$.

So, where is $rx$ for some $r \in R$?

Answer 1

$$rx\in Rx = \langle x\rangle$$ (In this notation, $\langle p(x)\rangle = R\cdot p(x)$)