Definition:Unital Subalgebra

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:

$(1) \quad B_R$ is unital

$(2) \quad$ Its unit is $1_A$.

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.