I was trying to solve this question from Artin:
Exercise 11.6.1. Let $φ : R[x] \rightarrow C\times C$ be the homomorphism defined by $φ(x) = (1, i)$ and $φ(r) = (r, r)$ for $r ∈ R$. Determine the kernel and the image of $φ$.
This is what the first part of the solution says: $φ$ is the evaluation map $f\mapsto(f (1), f (i))$. So $f \in \ker(\phi) \Leftrightarrow f(1) = f (i) = 0 \Leftrightarrow f \in ((x − 1)(x^2 + 1))$, i.e., $\ker(\phi) = ((x − 1)(x^2 + 1))$.
I have not come across this terminology yet and do not understand it. What is $f$ and where did it come from? Whether or not in this context, please can someone explain what an 'evaluation map' is?
It’s just that $f$ is a typical (“generic”) element of the domain, $R[x]$. An evaluation map generally is something that takes as its input a function and evaluates it at a previously given point.
Typically, you have a space, $\mathfrak X$, and a family $\mathfrak F$ of functions on $\mathfrak X$, say all with values in a ring $R$. Then for $x\in\mathfrak X$, you have the evaluation map $\text{ev}_x:\mathfrak F\to R$, defined by: for every $f\in\mathfrak F$, $\text{ev}_x(f)=f(x)$.