Subring Containing Ring Unity has Unity

Definition

Let $\struct{R, +, *}$ be a ring with unity $1_R$.

Let $\struct{S, +_S, *_S}$ be a subring of $\struct{R, +, *}$.

Let $1_R \in S$.

Then $1_R$ is the unity of $\struct{S, +_S, *_S}$.

Proof

We have:

\(\ds \forall s \in S: \, \)

\(\ds s *_S 1_R\)

\(=\)

\(\ds s * 1_R\)

Definition of Restriction of Operation

\(\ds \)

\(=\)

\(\ds s\)

Definition of Unity of Ring

\(\ds \)

\(=\)

\(\ds 1_R * s\)

Definition of Unity of Ring

\(\ds \)

\(=\)

\(\ds 1_R *_S s\)

Definition of Restriction of Operation

$\blacksquare$

Also see

• Unity of Subring is not necessarily Unity of Ring