nLab topological ring

Definition

Definition

A topological ring is a ring internal to the category Top of topological spaces, hence: a ring object in Top:

a topological space $R$ equipped with the structure of a ring on its underlying set, such that addition $+$ and multiplication $\cdot$ are continuous functions.

Definition

A topological field is a topological ring $K$ whose underlying ring is in fact a field and such that reciprocation $(-)^{-1}: K \setminus \{0\} \to K \setminus \{0\}$ is continuous. This latter condition is the same as demanding that the subspace topology on $K \setminus \{0\}$ induced by the embedding $K \setminus \{0\} \hookrightarrow K$ coincide with the subspace topology induced by the embedding $K \setminus \{0\} \to K \times K: x \mapsto (x, x^{-1})$. More at topological field.

Definition

A topological algebra — namely a topological associative algebra —, over a topological ring $R$ is a topological ring $S$ together with a topological ring homomorphism $R \to S$ that makes $S$ an $R$-algebra on the underlying set.

Properties

• The structure of a topological ring $(R,+,\cdot)$ makes $R$ a uniform space.

• It is automatic that negation in a topological unital ring is continuous, since it is the operation of multiplication with $-1$, and multiplication is continuous in each argument. Hence for $(R,+,\cdot)$ a topological ring, then $(R,+)$ is a topological group.

• In a topological ring, the closure of $\{0\}$ is an ideal. It follows that for a topological field $F$, either $0$ is a closed point (so that $F$ is $T_1$ and therefore completely regular Hausdorff, by standard arguments in the theory of uniform spaces), or is a codiscrete space.

Examples

• The real numbers and complex numbers form topological rings, in fact topological fields.

• Any pseudocompact ring such as the completed group ring of a profinite group is a topological ring.

• For any prime number $p$, the ring of p-adic integers is a topological ring.

• A Banach algebra is in particular a topological algebra, hence a topological ring. Hence so is a C-star-algebra.

Related concepts

• topological monoid

• normed ring

• local field

• completion of a ring

• linear topological ring

• real homotopy theory

• topological vector space

References

Lecture notes:

• Daniel Murfet, Topological rings (2005) [pdf]

See also:

• eom: Topological ring

• Stacks Project: Topological ring

• Wikipedia: Topological ring