ac.commutative algebra - What is the localization of Q[x]/(x) at 0

If you localize Q[x] at zero, you get the field of rational functions Q(x), so your module localizes to a vector space over this field (see Wikipedia). Your module is finitely generated, in fact by a single element, so you get a vector space of dimension zero or one. To decide which is correct, we can use Proposition 2.1 in Eisenbud's Commutative algebra (with a view toward algebraic geometry): If M is finitely generated, then $M[U^{-1}] = 0$ if and only if M is annihilated by an element of U. In this case, x is such an annihilating element, and your localized module is zero. Therefore, your first answer is correct, and your second answer is wrong.