Category:Unital Subalgebras

This category contains results about Unital Subalgebras.
Definitions specific to this category can be found in Definitions/Unital Subalgebras.

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 $A_R$.