This is a minor variation of MH's response:

A ring R is Boolean if x^2 = x for all x in R. (This implies R is commutative.)

In a Boolean ring R, every prime ideal is maximal, Moreover, the only local Boolean ring is Z/2Z. Therefore, R' := inverse limit_{p \in Spec R} R_p = (Z/2Z)^{# Spec R}. In particular, R' is either finite or uncountably infinite.

But there are certainly countably infinite Boolean rings (a fancy justification for this is the Lowenheim-Skolem theorem in model theory): take an uncountable Boolean ring, and consider the subring generated by a countably infinite set of generators.

For more details on Boolean rings, see e.g. Section 4.5 of

http://math.uga.edu/~pete/integral.pdf

This is a minor variation of MH's response:

A ring R is Boolean if x^2 = x for all x in R. (This implies R is commutative.)

In a Boolean ring R, every prime ideal is maximal, Moreover, the only local Boolean ring is Z/2Z. Therefore, R' := inverse limit_{p \in Spec R} R_p = (Z/2Z)^{# Spec R}. In particular, R' is either finite or uncountably infinite.

But there are certainly countably infinite Boolean rings (a fancy justification for this is the Lowenheim-Skolem theorem in model theory): take an uncountable Boolean ring, and consider the subring generated by a countably infinite set of generators.

For more details on Boolean rings, see e.g. Section 4.5 of

http://alpha.math.uga.edu/~pete/integral.pdf