Unitization of Algebra over Field is Unital Algebra over Field

Theorem
Let $K$ be a field.

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

Let $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ be the unitization of $A$.

Then $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ is a unital algebra over $K$.

Proof
From Direct Product of Vector Spaces is Vector Space, $\struct {A_+, +_{A_+}, \cdot_{A_+} }_K = \struct {A \times K, +_{A \times K}, \cdot_{A \times K} }_K$ is a vector space over $K$.

We show that $\circ_{A_+} : A_+ \times A_+ \to A_+$ is a bilinear mapping.

Let $\tuple {u, \alpha} \in A_+$, $\tuple {v, \beta} \in A_+$ and $\tuple {w, \gamma} \in A_+$.

Let $\lambda \in K$.

We have:

and:

So $\circ_{A_+}$ is bilinear.

So $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ is an algebra.

We show that $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ is unital.

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

Then, we have:

and:

So $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ is a unital algebra over $K$.