ring of continuous functions
Let $X$ be a topological space^{} and $C(X)$ be the function space consisting of all continuous functions^{} from $X$ into $\mathbb{R}$, the reals (with the usual metric topology^{}).
Ring Structure on $C(X)$
To formally define $C(X)$ as a ring, we take a step backward, and look at ${\mathbb{R}}^{X}$, the set of all functions from $X$ to $\mathbb{R}$. We will define a ring structure^{} on ${\mathbb{R}}^{X}$ so that $C(X)$ inherits that structure and forms a ring itself.
For any $f,g\in {\mathbb{R}}^{X}$ and any $r\in \mathbb{R}$, we define the following operations^{}:

1.
(addition) $(f+g)(x):=f(x)+g(x)$,

2.
(multiplication) $(fg)(x):=f(x)g(x)$,

3.
(identities^{}) Define $r(x):=r$ for all $x\in X$. These are the constant functions. The special constant functions $1(x)$ and $0(x)$ are the multiplicative and additive identities in ${\mathbb{R}}^{X}$.

4.
(additive inverse) $(f)(x):=(f(x))$,

5.
(multiplicative inverse^{}) if $f(x)\ne 0$ for all $x\in X$, then we may define the multiplicative inverse of $f$, written ${f}^{1}$ by
$${f}^{1}(x):=\frac{1}{f(x)}.$$ This is not to be confused with the functional^{} inverse^{} of $f$.
All the ring axioms are easily verified. So ${\mathbb{R}}^{X}$ is a ring, and actually a commutative ring. It is immediate that any constant function other than the additive identity is invertible.
Since $C(X)$ is closed under all of the above operations, and that $0,1\in C(X)$, $C(X)$ is a subring of ${\mathbb{R}}^{X}$, and is called the ring of continuous functions over $X$.
Additional Structures on $C(X)$
${\mathbb{R}}^{X}$ becomes an $\mathbb{R}$algebra if we define scalar multiplication by $(rf)(x):=r(f(x))$. As a result, $C(X)$ is a subalgebra^{} of ${\mathbb{R}}^{X}$.
In addition to having a ring structure, ${\mathbb{R}}^{X}$ also has a natural order structure, with the partial order defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$. The positive cone^{} is the set $\{f\mid 0\le f\}$. The absolute value^{}, given by $f(x):=f(x)$, is an operator mapping ${\mathbb{R}}^{X}$ onto its positive cone. With the absolute value operator defined, we can put a lattice^{} structure (http://planetmath.org/Lattice) on ${\mathbb{R}}^{X}$ as well:

•
(meet) $f\vee g:={2}^{1}(f+g+fg)$. Here, ${2}^{1}$ is the constant function valued at $\frac{1}{2}$ (also as the multiplicative inverse of the constant function $2$).

•
(join) $f\wedge g:=f+g(f\vee g)$.
Since taking the absolute value of a continuous function is again continuous, $C(X)$ is a sublattice of ${\mathbb{R}}^{X}$. As a result, we may consider $C(X)$ as a latticeordered ring of continuous functions.
Remarks. Any subring of $C(X)$ is called a ring of continuous functions over $X$. This subring may or may not be a sublattice of $C(X)$. Other than $C(X)$, the two commonly used latticeordered subrings of $C(X)$ are

•
${C}^{*}(X)$, the subset of $C(X)$ consisting of all bounded^{} continuous functions. It is easy to see that ${C}^{*}(X)$ is closed under all of the algebraic operations (ringtheoretic or latticetheoretic). So ${C}^{*}(X)$ is a latticeordered subring of $C(X)$. When $X$ is pseudocompact, and in particular, when $X$ is compact^{}, ${C}^{*}(X)=C(X)$.
In this subring, there is a natural norm that can be defined:
$$\parallel f\parallel :=\underset{x\in X}{sup}f(x)=inf\{r\in \mathbb{R}\mid f\le r\}.$$ Routine verifications show that $\parallel fg\parallel \le \parallel f\parallel \parallel g\parallel $, so that ${C}^{*}(X)$ becomes a normed ring^{}.

•
The subset of ${C}^{*}(X)$ consisting of all constant functions. This is isomorphic^{} to $\mathbb{R}$, and is often identified as such, so that $\mathbb{R}$ is considered as a latticeordered subring of $C(X)$.
References
 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title  ring of continuous functions 

Canonical name  RingOfContinuousFunctions 
Date of creation  20130322 16:54:54 
Last modified on  20130322 16:54:54 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  14 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54C40 
Classification  msc 54C35 