Definition:Unital Subalgebra
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Definition
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 $A_R$.
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 $A_R$.
Warning
It is possible for $B_R \subseteq A_R$ to be a subalgebra of $A_R$ such that it has a unit $1_B$ such that $1_B \ne 1_A$.
Such a subalgebra is a unital algebra, but is not a unital subalgebra of $A_R$.
Also see
- Equivalence of Definitions of Unital Subalgebra
- Example of Subalgebra which is Unital but not Unital Subalgebra
- Definition:Unital Subring
- Results about unital subalgebras can be found here.