Unitization of Algebra over Field is Unital Algebra over Field

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Theorem

Let $K$ be a field.

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

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:

\(\ds \paren {\tuple {u, \alpha} + \lambda \cdot_{A_+} \tuple {v, \beta} } \circ_{A_+} \tuple {w, \gamma}\) \(=\) \(\ds \tuple {u + \lambda v, \alpha + \lambda \beta} \circ_{A_+} \tuple {w, \gamma}\)
\(\ds \) \(=\) \(\ds \tuple {\paren {u + \lambda v} w + \gamma \paren {u + \lambda v} + \paren {\alpha + \lambda \beta} w, \paren {\alpha + \lambda \beta} \gamma}\) Definition of Unitization of Algebra over Field
\(\ds \) \(=\) \(\ds \tuple {u w + \gamma u + \alpha w + \lambda \paren {v w + \gamma v + \beta w}, \alpha \gamma + \lambda \beta \gamma}\)
\(\ds \) \(=\) \(\ds \tuple {u w + \gamma u + \alpha w, \alpha \gamma} +_{A_+} \lambda \tuple {v w + \gamma v + \beta w, \beta \gamma}\)
\(\ds \) \(=\) \(\ds \paren {\tuple {u, \alpha} \circ_{A_+} \tuple {w, \gamma} } +_{A_+} \paren {\lambda \cdot_{A_+} \paren {\tuple {v, \beta} \circ_{A_+} \tuple {w, \gamma} } }\) Definition of Unitization of Algebra over Field

and:

\(\ds \tuple {u, \alpha} \circ_{A_+} \paren {\tuple {v, \beta} + \lambda \circ_{A_+} \tuple {w, \gamma} }\) \(=\) \(\ds \tuple {u, \alpha} \circ_{A_+} \tuple {v + \lambda w, \beta + \lambda \gamma}\)
\(\ds \) \(=\) \(\ds \tuple {u \paren {v + \lambda w} + \alpha \paren {v + \lambda w} + \paren {\beta + \lambda \gamma} u, \alpha \paren {\beta + \lambda \gamma} }\) Definition of Unitization of Algebra over Field
\(\ds \) \(=\) \(\ds \tuple {u v + \alpha v + \beta u + \lambda \paren {u w + \alpha w + \gamma u}, \alpha \beta + \alpha \lambda \gamma}\)
\(\ds \) \(=\) \(\ds \tuple {u v + \alpha v + \beta u, \alpha \beta} +_{A_+} \lambda \tuple {u w + \alpha w + \gamma u, \alpha \gamma}\)
\(\ds \) \(=\) \(\ds \paren {\tuple {u, \alpha} \circ_{A_+} \tuple {v, \beta} } +_{A_+} \paren {\lambda \cdot_{A_+} \paren {\tuple {u, \alpha} \circ_{A_+} \tuple {w, \gamma} } }\) Definition of Unitization of Algebra over Field

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:

\(\ds \tuple { {\mathbf 0}_A, 1_K} \circ_{A_+} \tuple {x, \lambda}\) \(=\) \(\ds \tuple { {\mathbf 0}_A x + 1_K x + \lambda {\mathbf 0}_A, 1_K \lambda}\) Definition of Unitization of Algebra over Field
\(\ds \) \(=\) \(\ds \tuple {x, \lambda}\)

and:

\(\ds \tuple {x, \lambda} \circ_{A_+} \tuple { {\mathbf 0}_A, 1_K}\) \(=\) \(\ds \tuple {x {\mathbf 0}_A + 1_K x + \lambda {\mathbf 0}_A, 1_K \lambda}\) Definition of Unitization of Algebra over Field
\(\ds \) \(=\) \(\ds \tuple {x, \lambda}\)

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

$\blacksquare$

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