Definition:Topological Ring

This page is about Topological Rings in the context of topology. For other uses, see Definition:Ring.

Definition

Let $\struct{R, +, \circ}$ be a ring.

Let $\tau$ be a topology over $R$.

Then $\struct{R,+,\circ,\tau}$ is a topological ring if and only if:

$(1): \quad \struct{R, +, \tau}$ is a topological group

$(2): \quad \struct{R, \circ,\tau}$ is a topological semigroup.

Also see

• Definition:Topological Semigroup
• Definition:Topological Group
• Definition:Topological Field

• Criterion for Ring with Unity to be Topological Ring

• Results about topological rings can be found here.