One-point schemes are making me confused Let $X$ be the spectrum of the dual numbers $k[x]/(x^2)$ over a base field $k$.
a. Can we call such a scheme a one-point scheme? The reason why I find this confusing is the fact that whenever we look at the $S$-rational points of $X$, we end up with a multitude of rational points coming from varying $S$.
b. Is it correct to say that the image of a morphism of schemes $X\to Y$ is also a one-point scheme? The image of a singleton should clearly be a singleton, but what about $S$-points?
 A: a. Yes, the ring $k[x]/(x^2)$ has one prime ideal, $(x)$, so the scheme $\operatorname{Spec} k[x]/(x^2)$ has one point: the underlying topological space of $\operatorname{Spec} R$ is the set of prime ideals of $R$ with the topology that for each ideal $I\subset R$, the set of prime ideals containing $I$ is declared to be a closed subset.
b. Yes, but it's important to aware that the notion of "image" for schemes can behave in slightly unexpected ways and to check in on this stuff at some point in your education. (Search "scheme-theoretic image" for more details, but know that you don't actually have to worry when computing the scheme-theoretic image of a one-point scheme.)
The key aspect of your confusion with $S$-points is that when people say "this scheme has one point", they're talking about honest actual points, not $S$-points. If someone wanted to say "this scheme has only one $S$-point", they'll say that. It's rare to talk about all the $S$-points for all $S$ at the same time, and by that metric there is only one scheme with one $S$-point for all $S$: $\operatorname{Spec} \Bbb Z$, the final object in the category of schemes.
