Why $f(x) = 5x$ is not a ring homomorphism?

Question

Why the function $f: \mathbb Z \to \mathbb Z$ given by $f(x) = 5x$ is not a ring homorphism, since $f(a+b) = 5(a+b) = 5a + 5b = f(a) + f(b)$, and same for $f(a \cdot b)$.

Answer 1

Since $\mathbb{Z}$ is a ring, to be a homomorphism must have $f(a+b)=f(a)+f(b)$, and $f(ab)=f(a)f(b)$.
But $30=f(6)=f(2\cdot 3) \neq f(2)\cdot f(3)=10\cdot 15=150$.