Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 1

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Theorem

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


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 $I_{_R} : R \to R$ 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:

$\forall z \in R: \map {\paren{I_{_R} * y}} z = \map {I_{_R}} z * y$


Then:

$\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$


Proof

Let $y \in R$.

\(\ds \forall z \in R: \, \) \(\ds \map {\paren {y * I_{_R} } } z\) \(=\) \(\ds y * \map {I_{_R} } z\) Definition of $y * I_{_R}$
\(\ds \) \(=\) \(\ds y * z\) Definition of Identity Mapping $I_{_R}$
\(\ds \) \(=\) \(\ds \map {\lambda_y} z\) Definition of Left Regular Representation $\lambda_y$

From Equality of Mappings:

$\lambda_y = y * I_{_R}$


\(\ds \forall z \in R: \, \) \(\ds \map {\paren {I_{_R} * y} } z\) \(=\) \(\ds \map {I_{_R} } z * y\) Definition of $I_{_R} * y$
\(\ds \) \(=\) \(\ds z * y\) Definition of Identity Mapping $I_{_R}$
\(\ds \) \(=\) \(\ds \map {\rho_y} z\) Definition of Right Regular Representation $\rho_y$

From Equality of Mappings:

$\rho_y = I_{_R} * y$

$\blacksquare$