Showing that functions in the real numbers form rings under pointwise addition and multiplication Let $F(\mathbb{R})$ be the set of all functions $f : \mathbb{R} → \mathbb{R}$. Define pointwise addition and multiplication as follows. For any $f$ and $g$ in $F(\mathbb{R})$ let:
(i) $(f + g)(s) = f(x) + g(x)$ for all $x \in \mathbb{R}$
(ii) $(f · g)(s) = f(x) · g(x)$ for all $x \in \mathbb{R}$
Prove that $F(\mathbb{R})$ forms a ring under these two operations.
I know the axioms of rings that I need to show hold for this to be true. My question is, when I am showing closure for instance,
$(f + g)(s) +(f + g)(t) = f(x) + g(x)+ f(y) + g(y) \in \mathbb{R}$ and
$(f · g)(s)·(f · g)(t) = f(x) · g(x) · f(y) · g(y) \in \mathbb{R}$
is this the correct way to apply the axioms? My book was very general about the definition of a ring. The whole pointwise notation versus the definition is confusing me. Any help would be appreciated.
 A: For closure of addition, since for any $x\in\Bbb{R}$, $f(x)+g(x)\in \Bbb{R}$, $f(x)+g(x)$ is well defined and so $(f+g)(x)\in F(\Bbb{R})$. 
Likewise for closure of multiplication, since for any $x\in\Bbb{R}$, $f(x)\cdot g(x)\in \Bbb{R}$, $f(x)\cdot g(x)$ is well defined and so $(f\cdot g)(x)\in F(\Bbb{R})$. 
Clearly ring axiom like commutative, assiciative and distributive laws for addition and multiplication holds for $F(\Bbb{R})$ since real numbers hold such laws. 
For example to prove commutativity, for any $x\in\Bbb{R}$
$$
(f+g)(x)=f(x)+g(x)=g(x)+f(x)=(g+f)(x)
$$
To prove associativity, for any $x\in\Bbb{R}$
$$
(f+(g+h))(x)=f(x)+(g(x)+h(x))=(f(x)+g(x))+h(x)=((f+g)+h)(x)
$$
$f(x)=0$ is the additive zero in $F(\Bbb{R})$. $f(x)=1$ is the multiplicative unity in $F(\Bbb{R})$. $-f(x)$ is the additive inverse for $f(x)$ in $F(\Bbb{R})$. 
A: More generally, if $X$ is a set and $R$ is a ring, then the set of all functions $X \to R$ with pointwise operations is a ring. This ring is commutative iff $R$ is commutative.
The proof of this more general fact should be exactly the same as the one you gave for $\mathbb R \to \mathbb R$.
It is useful to try to distill examples into their essential features. In your example, the fact that the domain of the functions is $\mathbb R$ is immaterial but the fact that the codomain is a ring is essential.
