Let us work over a field that is NOT algebraically closed. Specifically, $K=\mathbb R$.
We can think of the affine line $\mathbb A_K^1$ as a ringed space $(\mathbb A_K^1,\mathcal O_{\mathbb A^1_K})$, where for an open set $U\subseteq\mathbb A^1_K$, we have $\mathcal O_{\mathbb A^1_K}(U)=(\text{the ring of regular functions over $U$})=\{\frac fg\mid f,g\in K[x] \,\&\,(\forall P\in\mathbb A^1_K)(g(P)\ne 0)\}$.
I think that according to this, $\frac1{1+x^2}$ should be a regular function over the entire $\mathbb A^1_K$. Moreover, it seems to me that $f:\mathbb A_K^1\to\mathbb A_K^1$ defined as $f(x)=\frac1{1+x^2}$ is a morphism from the ringed space $(\mathbb A^1_K,\mathcal O_{\mathbb A^1_K})$ to itself (in the sense that it pulls back regular functions to regular functions). Am I right?
I am probably confused by either different sources using different definitions or me seeing statements that only hold for $K$ algebraically closed. E.g. the Wikipedia article Morphism of algebraic varieties states
The regular functions on $\boldsymbol A^n$ are exactly the polynomials in $n$ variables […]
,
which I only agree with for $K$ algebraically closed and not in the case I've shown above.
Best Answer
The definition of a regular map needs to be modified substantially when you work over a field that isn't algebraically closed. There are different ways to set things up at different levels of sophistication; one way to say it is that regular maps should have the property that they remain regular maps after extension of scalars, and your proposed map does not remain regular after extension of scalars to $\mathbb{C}$ because of the poles at $x = \pm i$.
You are implicitly thinking only in terms of what are called the $K$-points of a variety; this is only safe to do when $K$ is algebraically closed. For example if $K = \mathbb{F}_q$ is a finite field then the polynomial $f(x) = x^q - x$ vanishes on $K$-points but is not considered to be identically zero as a regular map, because it does not remain identically zero after extension of scalars, e.g. to the algebraic closure $\overline{\mathbb{F}_q}$. We also have issues such as varieties, for example $\text{Spec } \mathbb{R}[x, y]/(x^2 + y^2 + 1)$, which have no $K$-points (here $K = \mathbb{R}$) but which it would be incorrect to consider empty; here again the issue is that points exist after extension of scalars.
Here I am implicitly taking a functor of points approach which I personally find conceptually cleanest. The standard approach you will find in most textbooks is based on the Zariski spectrum of all prime ideals (which capture points over extensions of scalars but somewhat indirectly), not just maximal ideals corresponding to $K$-points.
On the other hand, there is a somewhat separate field of real algebraic geometry in which we only consider $\mathbb{R}$-points, and which is mostly not studied by algebraic geometers but by model theorists or something like that.