topological ring

A ring $R$ which is a topological space is called a topological ring if the addition, multiplication, and the additive inverse functions are continuous functions from $R\times R$ to $R$.

A topological division ring is a topological ring such that the multiplicative inverse function is continuous away from $0$. A topological field is a topological division ring that is a field.

Remark. It is easy to see that if $R$ contains the multiplicative identity $1$, then $R$ is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the product of the element and $-1$. However, if $R$ does not contain $1$, it is necessary to impose the continuity condition on the additive inverse operation.