Let $A$ be the ring $\mathbb{Z}/60$. What does it look like under the localization at the prime (2)? Everything that is not divisible by 2 should be a unit, so one might thing this looks like $\mathbb{Z}/4$ - but on the other hand, rings always inject into their localizations...
I'm asking because Vakil in his notes says that the stalk at (2) of this ring is $\mathbb{Z}/4$. But this doesn't make sense to me because of the above confusion.