Premise: I'm not a mathematician, please be patient.
Anyway, if I consider the Galois field $GF(2)$ and two operations $+$ (inclusive or, also denoted with $\lor$) and $\neg$ (negation), where, given $x,y \in GF(2)$: $$ x + y := (x \cdot y) \oplus (x \oplus y), $$ $$ \neg x:= x \oplus 1, $$ I've defined a boolean ring $(GF(2),+,\neg)$.
Note: I know that in information theory, code theory and cryptography a bit is usually defined in $GF(2)$, and in these cases there's no need to use Boolean rings, but for example in some context in computer science or digital electronics you use boolean algebra, since you also need inclusive or and "negated" variables or signals.
Does this definition work in both cases? Is it safe to say that mathematically bits are defined in a Boolean ring?
Another question is: if I need a string of $n$-bit $ \boldsymbol{b} = [b_1, b_2, \dots, b_n], $ is it safe to say that $\boldsymbol{b} \in (GF(2^n),+,\neg)$?
$GF(2)$ forms a boolean ring. It follows that $GF(2)^n$ forms a boolean ring, because the operations are defined row by row and the rows are independently considered (as in linear algebra).