Understanding the proof of the existence of absolute integral closure

Question

I'm currently reading this proof in the stacks project showing the existence of absolutely integral closure. Here, we consider the set $I$ of all monic polynomials with coefficients in $A$ (so an element of $I$ is in $A[x]$). Then we construct a new ring $A[x_i;i \in I]/(P_i;i \in I)$, where $P_i$ is the corresponding polynomial in the variable $x_i$. The part I cannot understand is that the canoniacal map $A \to A[x_i;i \in I]/(P_i;i \in I)$ is an embedding. This is equivalent to saying that $A \cap (P_i;i \in I) = 0$, but how can we rigorously show that any linear combination of these polynomials does not magically cancel each other and only leave the constant term?