Convergence properties in a Hausdorff space that is a ring

Question

Suppose $T$ is a Hausdorff space and that $T$ forms a ring with identity.

Is the following claim true?

If $lim_{n\to \infty}a_n=0$,

Then $lim_{n\to \infty}ra_n=0$ .

(where $a_n \in T$ and $r \in T$)

Answer 1

For $T$ a topological ring the function $F:T \times T \to T$ given by $F(x,y) = y$ is continuous by definition. For fixed $r \in T$ and the restriction $F: \{r\} \times T \to T$ is continuous. From this it is an exercise on the product topology to show $G:T \to T$ given by $G(x) = rx$ is continuous and your claim about sequences holds.