Equivalence of Definitions of Unital Subalgebra
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Theorem
Let $R$ be a commutative ring.
Let $\struct {A_R, *}$ be an unital algebra over $R$ whose unit is $1_A$.
Let $\struct {B_R, *}$ be a subalgebra of $A_R$.
The following definitions of the concept of Unital Subalgebra are equivalent:
Definition 1
$\struct {B_R, *}$ is a unital subalgebra of $A_R$ if and only if $1_A \in B$.
Definition 2
$\struct {B_R, *}$ is a unital subalgebra of $A_R$ if and only if:
That is, a unital subalgebra of $A_R$ must not only have a unit, but that unit must also be the same unit as that of $A_R$.
Proof
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