External Direct Product Identity/Sufficient Condition

Theorem
Let $\struct {S \times T, \circ}$ be the external direct product of two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.

Let:
 * $\struct {S, \circ_1}$ have identity element $e_S$

and:
 * $\struct {T, \circ_2}$ have identity element $e_T$.

Then $\tuple {e_S, e_T}$ is the identity element for $\struct {S \times T, \circ}$.

Proof
Let $e_S$ and $e_T$ be the identity elements of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ respectively.

Thus $\tuple {e_S, e_T}$ is the identity of $\struct {S \times T, \circ}$.

Also see

 * External Direct Product Associativity
 * External Direct Product Commutativity
 * External Direct Product Inverses