$F[x]$ is a ring

The properties of addition and multiplication are summarized in the following proposition.

Proposition 2.2.3   Let $F$ be a field or one of the rings $\mathbb{Z}$ or $\mathbb{Z}/m\mathbb{Z}$. For any polynomials $a(x),b(x),c(x)\in F[x]$,

1. $a(x)+b(x)=b(x)+a(x)$, (``addition is commutative'')
2. $a(x)b(x)=b(x)a(x)$, (``multiplication is commutative'')
3. $(a(x)+b(x))+c(x) =a(x)+(b(x)+c(x))$, (``addition is associative'')
4. $(a(x)b(x))c(x) =a(x)(b(x)c(x))$, (``multiplication is associative'')
5. $(a(x)+b(x))c(x)= a(x)c(x)+b(x)c(x)$, (``distributive'')
6. $a(x)\cdot 1= a(x)$, (``$1$ is a multiplicative identity'')
7. $a(x)+(-a(x))=0$,
8. $a(x)\cdot 0=0$,
9. $a(x)+0=a(x)$ (``0 is a additive identity''),
10. if $a(x)c(x)=b(x)c(x)$ and $c(x)\not= 0$ then $a(x)=b(x)$ (``cancellation law'').

In other words, $F[x]$ is a ring.