In another, now deleted, venue you mentioned a more specific question of just showing the operations are continuous. You mentioned you could handle negation, but had trouble with addition and multiplication. Here is one way to verify them:
Let J = In.
The inverse image of x+J under addition is { (y,z) : y+z-x in J }. This is a union of A(y) = (y+J)⊕(x-y+J) ≤ R⊕R as y varies over R. Each A(y) is open (being a direct product of open sets), and so the union is open. In other words, the preimage under + of an open set is open.
The inverse image x+J under multiplication is { (y,z) : yz-x in J }. This is a union of B(y) = (y+J)⊕(∪{ z+J : yz-x in J }). Each B(y) is open (being a direct product of an open set and a union of open sets). In other words the preimage under multiplication of an open set is open.
The only trick to it is (1) knowing some open sets of a direct product and (2) noticing that you could have just worked in a quotient ring R/J where every set (every!) is open, since it is a union of singletons, x+J. In plainer language, (y+j)*(z+j') = yz + jz+yj'+jj' = yz + (something in J) is still in yz+J.