Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms
Theorem
Let $\struct{R, + , \circ, \tau}$ be a topological ring with unity $1_R$.
For all $y \in R$, let $\lambda_y$ and $\rho_y$ denote the left and right regular representations of $\struct{R, \circ}$ with respect to $y$.
Let $x \in R$ be a unit of $R$ with product inverse $x^{-1}$.
Then $\lambda_x, \, \rho_x: \struct{R, \tau} \to \struct{R, \tau}$ are homeomorphisms with inverse mappings $\lambda_{x^{-1} }, \, \rho_{x^{-1} }: \struct{R, \tau} \to \struct{R, \tau}$ respectively.
Proof
Let $I_{_R} : R \to R$ be the identity mapping on $R$.
For all $y \in R$, let $y * I_{_R} : R \to R$ be the mapping defined by:
- $\forall z \in R: \map {\paren {y * I_{_R} } } z = y * \map {I_{_R}} z$
For all $y \in R$, let $I_{_R} * y : R \to R$ be the mapping defined by:
- $\forall z \in R: \map {\paren {I_{_R} * y}} z = \map {I_{_R}} z * y$
Lemma 1
- $\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$
$\Box$
From Identity Mapping is Continuous, $I_{_R} : \struct{R, \tau} \to \struct{R, \tau}$ is continuous.
From Multiple Rule for Continuous Mappings into Topological Ring:
- $x * I_{_R}, \, I_{_R} * x, x^{-1} * I_{_R}, \,I_{_R} * x^{-1}$ are continuous.
Lemma 2
- $x * I_R$ is a bijection and $x^{-1} * I_R$ is the inverse of $x * I_R$
- $I_R * x$ is a bijection and $I_R * x^{-1}$ is the inverse of $I_R * x$
$\Box$
By definition of homeomorphism, $\lambda_x, \,\rho_x : \struct{R, \tau} \to \struct{R, \tau}$ are homeomorphisms with inverse mappings $\lambda_{x^{-1} }, \,\rho_{x^{-1} } : \struct{R, \tau} \to \struct{R, \tau}$ respectively.
$\blacksquare$