Subring of Commutative Ring is Commutative

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

$\blacksquare$