Subring Containing Ring Unity has Unity
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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 |
$\blacksquare$