Subring of Commutative Ring is Commutative
Jump to navigation
Jump to search
Definition
Let $\struct{R, +, *}$ be a commutatve ring.
Let $\struct{S, +_S, *_S}$ be a subring of $\struct{R, +, *}$.
Then:
- $\struct{S, +_S, *_S}$ is a commutatve ring
Proof
We have:
\(\ds \forall s, t \in S: \, \) | \(\ds s *_S t\) | \(=\) | \(\ds s * t\) | Definition of Restriction of Operation | ||||||||||
\(\ds \) | \(=\) | \(\ds t * s\) | Definition of Commutative Ring | |||||||||||
\(\ds \) | \(=\) | \(\ds t *_S s\) | Definition of Restriction of Operation |
$\blacksquare$