Algebra over Field Embeds into Unitization as Ideal

Theorem
Let $K$ be a field.

Let $A$ be an algebra over $K$ that is not unital.

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
From Algebra over Field Embeds into Unitization as Vector Subspace, $A_0$ is a vector subspace of $A_+$.

It remains to show that for each $u \in A_0$ and $v \in A_+$, we have $u v \in A_0$ and $v u \in A_0$.

Let $x \in A$.

Let $\tuple {y, \lambda} \in A_+$.

Then, we have:

and: