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:
- $\rho_y = I_{_R} * y$
$\blacksquare$