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

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## 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.

## Facts

### Facts about the max-spectrum

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

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.