Additive Regular Representations of Topological Ring are Homeomorphisms

Theorem

Let $\struct {R, + , \circ, \tau}$ be a topological ring.

Let $x \in R$.

Let $\lambda_x$ and $\rho_x$ be the left and right regular representations of $\struct {R, +}$ with respect to $x$.

Then $\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are homeomorphisms with inverses $\lambda_{-x}, \,\rho_{-x}: \struct {R, \tau} \to \struct {R, \tau}$ respectively.

Proof

By definition of a topological ring, $ \struct {R, + , \tau}$ is a topological group.

From Right and Left Regular Representations in Topological Group are Homeomorphisms:

$\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are homeomorphisms with inverses $\,\lambda_{-x}, \,\rho_{-x}: \struct {R, \tau} \to \struct {R, \tau}$ respectively.

$\blacksquare$