# Definition:Completion (Normed Division Ring)

## Definition

Let $\struct {R_1,\norm{\,\cdot\,}_1}$ and $\struct {R_2,\norm{\,\cdot\,}_2}$ be normed division rings.

Let $M_1 = \left({R_1, d_1}\right)$ and $M_2 = \left({R_2, d_2}\right)$ be the metric spaces where $d_1: R_1 \times R_1 \to \R_{\ge 0}$ and $d_2: R_2 \times R_2 \to \R_{\ge 0}$ are the metrics induced by $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ respectively.

Then $\struct {R_2,\norm{\,\cdot\,}_2}$ is a completion of $\struct {R_1,\norm{\,\cdot\,}_1}$, or $\struct {R_2,\norm{\,\cdot\,}_2}$ completes $\struct {R_1,\norm{\,\cdot\,}_1}$, if and only if:

$(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$.
$(2): \quad M_2$ is a metric completion of $\phi \paren{M_1}$.

That is, $\struct {R_2,\norm{\,\cdot\,}_2}$ is a completion of $\struct {R_1,\norm{\,\cdot\,}_1}$ if and only if

$(a): \quad M_2$ is a complete metric space.
$(b): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$.
$(c): \quad \phi \paren{R_1}$ is a dense subspace in $M_2$.