Affine line

Let $k$ be a field. The affine line over $k$ is the affine scheme

where $k[x]$ is the polynomial ring in one variable. Its points are all prime ideals of $k[x]$, not only elements of $k$.

Each $a\in k$ determines the closed point $(x-a)$. If $k$ is algebraically closed, these are exactly the closed points, while $(0)$ is a generic point whose closure is all of $\mathbb A_k^1$. Thus the scheme-theoretic affine line contains the familiar coordinate line together with extra information about irreducibility and specialization.

For example, the basic Zariski open subset $D(x)$ removes the origin. Its ring of regular functions is $k[x,x^{-1}]$, so $x$ becomes invertible there. The higher-dimensional version is affine $n$-space.