Problem:
Show that the following maps are (not) ring homomorphisms (with or without 1):
$\phi: \mathbb{C} \to \mathbb{R}^{2\times2}, a +bi \mapsto \left( \begin{array}{rrr}a & b\\ -b & a \\ \end{array}\right)$
$inv: \mathbb{C} \to \mathbb{C}, a+bi \mapsto (a+bi)^{-1}$
$pot_2 : \mathbb{C} \to \mathbb{C}, a+bi \mapsto (a+bi)^2$
$\phi_i: \mathbb{R}[X] \to \mathbb{C}, f \mapsto f(i)$ ($i$ being the imaginary number)
Questions:
I would like to know more about how to figure out that these are ring homomorphisms, as i am not really well versed in the complex numbers. Would there be any sort of advice on the matter ? I appreciate any help given.