Additive Regular Representations of Topological Ring are Homeomorphisms

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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.

$\blacksquare$