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

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 $y \in R$.

From Equality of Mappings:
 * $\lambda_y = y * I_{_R}$

From Equality of Mappings:
 * $\rho_y = I_{_R} * y$