(Commutative algebra and algebraic geometry - Bosch - page 16)
I am struggling to reconcile the highlighted items in the following extract:
I can understand (2) : if Ring element $f$ is a member of the prime ideal $\mathfrak{p}_x$ where $x$ is an index in the list of prime ideals for the Ring, then we set $f(x) \neq "0"$ where (I am guessing) $"0"$ is the $0$ element of the Ring
But (with that guess) I cannot understand how this definition of $f(x)$ reconciles with the co-domain of $f$ as stated in (1) : there, the co-domain is stated as a disjoint union of sets of cosets, 1 set of coset per each $\mathfrak{p}_x$.
How does $"0"$ fit in that co-domain ?
There are many zeros, one for each term in the union. So for instance $f(x) = 0$ really means $f(x) = 0_x$ Where $0_x\in R/p_x$ and $0_x \neq 0_y$ when $x\neq y$.