Topological ring

In mathematics, a topological ring is a ring ${\displaystyle R}$ that is also a topological space such that both the addition and the multiplication are continuous as maps^[1] ${\displaystyle R\times R\to R}$ where ${\displaystyle R\times R}$ carries the product topology. This means ${\displaystyle R}$ is an additive topological group and a multiplicative topological semigroup.

Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example the completion of a topological field may be a topological ring which is not a field.^[2]