Algebra over Field Embeds into Unitization as Ideal

Theorem

Let $K$ be a field.

Let $A$ be an algebra over $K$.

Let $A_+$ be the unitization of $A$.

Let:

$A_0 = \set {\tuple {x, 0_K} : x \in A} \subseteq A_+$.

Then $A_0$ is an ideal in $A_+$.

Proof

This follows from Ideal of Algebra over Field Embeds into Unitization as Ideal, since $A$ is an ideal of itself.

$\blacksquare$