Intuition for a ring homomorphism? A map $f: A \to B$ between two rings $A$ and $B$, is called a ring homomorphism if $f(1_A) = 1_B$, and one has $f(a_1 + a_2) = f(a_1) + f(a_2)$ and $f(a_1 \cdot a_2) = f(a_1) \cdot f(a_2)$, for any $a_1, a_2 \in A$.
My question is, what is the intuition for ring homomorphisms? My grasp on them is kind of bad...

Is $482350923581014689 + 51248129432153$ an even or odd number? Of course it is even because adding two odd numbers gives an even number right? You didn't have to calculate.
Is $1254346425649847 \times 64341341232606942$ an even or odd number? Well, an odd number times an even number must be even. Again, you didn't have to multiply that big numbers.
Now consider the ring homomorphism $\varphi : \mathbb Z \to \mathbb Z_2$ given by
$$\varphi(x)= \begin{cases} 
      0 & \text{if $x$ is even} \\
      1 & \text{if $x$ is odd}
   \end{cases}
$$
In $\mathbb Z_2$, we have $1+1=0$ which means adding two odd numbers gives an even number. Also, $1 \cdot 0 =0$ and so, multiplication of an odd number and an even number gives an even number.
In short, making the operation first and finding its image is the same as finding the image and then making the operation in ring homomorphisms.
