Consider the following example:
On page 1 of my paper I write: Let $R$ be a finite commutative ring with $1$.
...[some text where I use $R$]...
On page 3 of the paper I write: $\textbf{Definition.}$ Let $R$ be a commutative ring with $1$ and let $U$ be a subset of $R$. The annihilator of $U$ is defined as $\mathrm{Ann}(U):= \{r\in R:rU=0\}.$
Now, consider $r\in R$. <- Would it be clear that $R$ is the finite ring from page 1, which I have used so far, or did the $\textbf{Definition}$ "overwrite" the meaning of $R$?
Is it instead preferable to give the commutative ring in the definition a different name, such as $R'$ for example?
P.S. What are common letters for writing rings apart from $R$ and $S$?