I'm trying to solve a homework exercise which starts off:
Let $k$ be a field, and let $A = k[x_1, x_2, \dots]$ be a polynomial ring in infinitely many variables. Let $\mathfrak p \subseteq A$ be the prime ideal $(x_1, x_2, \dots)$. Let $M$ be the quotient $A/(x_1^2 - x_1, x_2^2 - x_2, \dots)$ considered as an $A$-module. Show that the localization $M_\mathfrak p$ is isomorphic to $k$.
So I'm trying to reason about $M_\mathfrak p = S^{-1} M$ for $S = A \setminus \mathfrak p$. But isn't $S$ equal to $k$, since $\mathfrak p$ is composed of all polynomials of degree $\geq 1$? Wouldn't $M_\mathfrak p$ then just equal $M$ itself, which I think is composed of all polynomials in $x_1, x_2, \dots$ of degree $1$.
What am I doing wrong?