Embedding integers in the inverse limit

Question

Let $S=(R_m,f_{mn})_{m\geq n\geq 0}$ be an inverse system of polynomial rings over the integers and unital ring homomorphisms between them. Let $R_S$ be the inverse limit of $S$ in the category Ring of rings, with the projection maps $\pi_m:R_S \to R_m$ for all $m \geq 0$. I want to show that there is an embedding of $\mathbb{Z}$ in $R_S$. I argue as follows. Let $R$ be the canonical inverse limit of $S$. That is, \begin{equation} R=\bigg\{(x_m)_{m \geq 0} \in \displaystyle \prod_{m \geq 0} R_m:f_{mn}(x_m)=x_n \; \forall \; m\geq n \geq 0 \bigg\}, \end{equation} equipped with canonical projections. We have that $(k)_{m \geq 3} \in R$ for all $k \in \mathbb{Z}$ and so there is a canonical embedding $f:\mathbb{Z} \to R$ which sends $k$ to $(k)_{m \geq 0}$, which is unique (since $\mathbb{Z}$ is a cone over $S$). Now, let $u:R_S \to R$ be the unique ring isomorphism from $R_S$ to $R$. Then $g:\mathbb{Z} \to R_S$, where $g=u^{-1}f$, is a unique embedding of $\mathbb{Z}$ in $R_S.$ Is this correct?