Mathematical Writing: Can objects in a definition be assigned any name, even though this name is used elsewhere in the text?

Question

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 $mathrm{Ann}(U):= {rin R:rU=0}.$
Now, consider $rin R$. <- 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 $S$?

Matt Samuel · Answer

If you define $R$ on page $1$ in some definition then use it later outside of the definition, the implication is that it is the same $R$. However, if you then in another definition say "Let $R$ be a commutative ring with $1$", then the implication is that it is independent of the original $R$ and generic, possibly the same but possibly different. If you want it to be different, you could say something like (not necessarily the $R$ used above). But you shouldn't do this if you intend to continue using the specific previous $R$.
Giving it a different name makes it clear what you intend, but it's not a great solution if you have to do this many times and have no intention of carrying the rings in previous definitions around. You'll run out of letters eventually. There are ways to make it clear without changing the name if you don't want to keep the old one around.

Mathematical Writing: Can objects in a definition be assigned any name, even though this name is used elsewhere in the text?

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