# Ring of continuous real-valued functions on a topological space

## Contents

## Definition

The **ring of continuous real-valued functions on a topological space** is defined as the ring of all continuous functions from the topological space to the real numbers (endowed with the usual topology), with pointwise addition and multiplication.

## Related constructions

- Ring of continuous complex-valued functions on a topological space
- Sheaf of continuous real-valued functions on a topological space

## Facts

### Facts about the max-spectrum

- Topological space maps naturally to max-spectrum of ring of continuous real-valued functions: For any topological space , there is a natural injective map from to the ring of continuous real-valued functions on . This map sends a point to the ideal of all continuous real-valued functions such that .
- Natural map from topological space to max-spectrum of ring of continuous real-valued functions is an injection iff the space is Urysohn: In case of a Urysohn space, the map defined above is an injection. Otherwise, it isn't.
- Natural map from topological space to max-spectrum of ring of continuous real-valued functions is a surjection iff the space is compact: In case of a compact space, the map defined above is a surjection. If the space is not compact, the map is not a surjection.
- Max-spectrum of ring of continuous real-valued functions on completely regular space contains homeomorphic copy of space: In case of a completely regular space, the map defined above is a homeomorphism to its image, given the subspace topology from the natural topology on the max-spectrum.
- Max-spectrum of ring of continuous real-valued functions on compact Hausdorff space is naturally homeomorphic to the space: This is a consequence of the previous two facts.
- Maximal ideal in ring of continuous real-valued functions on nontrivial compact Hausdoff space is not finitely generated: This in particular implies that such a ring is not a Noetherian ring.

Analogous statements hold for the ring of continuous complex-valued functions on a topological space.

## Ring properties

### Normality

*This ring is a normal ring*

The ring of continuous real-valued functions is a normal ring: it is integrally closed inside its total quotient ring. *For full proof, refer: Ring of continuous real-valued functions on a topological space is normal*

### Noetherianness

*This ring is not in general a Noetherian ring*

The ring of continuous real-valued functions on a topological space is not a Noetherian ring in general. However, the ring of continuous real-valued functions over a Noetherian space is a Noetherian ring. Over a completely regular space that is not Noetherian, the ring of continuous real-valued functions is not Noetherian.

### Reduced ring

'*This ring is a reduced ring*

The ring of continuous real-valued functions on a topological space is a reduced ring: it has no nonzero nilpotent elements.

## Element properties

### Zero divisors

`Further information: Classification of zero divisors in ring of continuous real-valued functions on a topological space`

Any zero divisor must have the property that its zero set contains a nonempty open subset. This condition is also sufficient for a completely regular space.