Ring homomorphism of polynomial ring 
Let $R$ be a ring, and $R[x]$ the polynomial ring.
$R\left [ x \right ]\rightarrow R$
$f\left [ x \right ] \mapsto f\left ( 0 \right )$
is a ring homomorphism

I'm having a bit of trouble verifying this is true. Would someone kindly provide some help?
Thanks in advance.
 A: Hint:
Evaluation at $0$ associates to every polynomial its constant term. So you have to check the constant term of $f(x)+g(x)$ is the sum of the constant terms of $f(x)$  and $g(x)$, ans similarly for their product.
A: In general, for any $r\in R$ the map $\phi: R[x]\rightarrow R$ defined $f\in R[x]\mapsto f(r)$ is a ring homomorphism.
To show this, we need to check that for any $f,g\in R[x]$, $(f+g)(r) = \phi(f+g) = \phi(f)+\phi(g) = f(r)+g(r)$, and $(fg)(r) = \phi(fg) = \phi(f)\phi(g) = f(r)g(r)$.
But the order in which we replace $x$ by $r$ doesn't matter. For example, write $f = a_{n}x^{n}+\cdots+a_{0}$ and $g = b_{m}x^{m}+\cdots+b_{0}$ so $fg = (a_{n}x^{n}+\cdots+a_{0})(b_{m}x^{m}+\cdots+b_{0}) = c_{k}x^{k}+\cdots+c_{0}$ for some $k$ and some $c_{i}\in R$.
Then $(fg)(r) = c_{k}r^{k}+\cdots+c_{0}$. But $(a_{n}x^{n}+\cdots+a_{0})(b_{m}x^{m}+\cdots+b_{0}) = c_{k}x^{k}+\cdots+c_{0}$, still holds when $x$ is replaced with any element of $R$. So in particular, $(a_{n}r^{n}+\cdots+a_{0})(b_{m}r^{m}+\cdots+b_{0}) = c_{k}r^{k}+\cdots+c_{0}$. And here $(a_{n}r^{n}+\cdots+a_{0})(b_{m}r^{m}+\cdots+b_{0}) = f(r)g(r)$.
Similarly $(f+g)(r) = f(r)+g(r)$.
