Definition:Topological Ring

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

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

Then $\left({R,+,\circ,\tau}\right)$ is a topological ring iff:


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


 * $(2): \quad \circ$ is continuous when considered as a mapping from $\left({R, \tau}\right) \times \left({R, \tau}\right)$ to $\left({R, \tau}\right)$.

Remark
If $\left({R,+,\circ}\right)$ is a Ring with Unity, we need only require that $+$ and $\circ$ are continuous, because for each $x \in R$, $-x = (-1_R) \circ x$.