Criterion for Ring with Unity to be Topological Ring

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

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

Suppose that $+$ and $\circ$ are $\tau$-continuous mappings.

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

Proof
As we presume $\circ$ to be continuous, we need only prove that $\left({R, +, \tau}\right)$ is a topological group.

As we presume $+$ to be continuous, we need only show that negation is continuous.

For each $b \in R$, $- b = (- 1_R) \circ b$.

Since $\circ$ is continuous, it is continuous in each argument, so negation is continuous.