Can a Boolean ring be somehow extended to infinite binary sequences? If $2^{\Bbb N}$ is the Cantor space (the set of all infinite binary sequences), is it a ring if the operations are defined term by term?
For example, given the sequences $A:=\{a_n\}_{n\ge1}$ and $B:=\{b_n\}_{n\ge1}$, $\:A+B\:$ and $AB\:$ would be $\{a_n+b_n\}_{n\ge1}$ $\{a_nb_n\}_{n\ge1}$ respectively
One of the operations is $AND$/conjunction but I don´t know if the other should be $OR$/disjunction or $XOR/$exclusive disjunction. I´m also not sure about inverses: one sequence can have many different "inverses" under the operation $AND$. Does this mean that the conjunction is the multiplication and all those inverses are the "divisors of zero"?
You have indeed two options. If you equip the set $\{0, 1\}$ with $OR$ and $AND$ you have a semiring structure. If you make use of $XOR$ and $AND$, you have a Boolean ring. If $S$ is one of these two structures, the product $S^\mathbb{N}$ has the same structure, defined componentwise as you did.