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.

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 $X$, there is a natural injective map from $X$ to the ring of continuous real-valued functions on $X$. This map sends a point $x\in X$ to the ideal $M_{x}$ of all continuous real-valued functions $f:X\to \mathbb {R}$ such that $f(x)=0$.
• 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.