Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms

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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.

$\blacksquare$

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