This pertains to Ex. 1.13 (self-studier) in Reid's "Undergrad. Commutative Algebra":
If $A$ is a reduced ring and has finitely many minimal prime ideals $P_i$ then
$A\hookrightarrow \bigoplus_{i=1}^n A/P_i$; moreover, the image has nonzero intersection with each summand.
I "know" how to solve the problem, showing that the kernel of the map is zero, and designating $a_i \notin P_i$ while in $\bigcap P_j$ with $j\not= i$.
This has naively ignored the stipulation of "a reduced ring," which is my question:
What would the presence of nilpotents in $A$ do? My guess is that it would make the kernel of the map non-trivial. If that is correct, I would please appreciate help in seeing how this comes about.
If that guess is not correct, then I would again appreciate even more help.
Thanks.