Ring of continuous real-valued functions on a topological space

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

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.