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.
$\blacksquare$