Theorem: Unitization of Algebra over Field is Unital Algebra over Field

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:
{{begin-eqn}}
{{eqn	| l = \paren {\tuple {u, \alpha} + \lambda \cdot_{A_+} \tuple {v, \beta} } \circ_{A_+} \tuple {w, \gamma}
		| r = \tuple {u + \lambda v, \alpha + \lambda \beta} \circ_{A_+} \tuple {w, \gamma}
}}
{{eqn	| r = \tuple {\paren {u + \lambda v} w + \gamma \paren {u + \lambda v} + \paren {\alpha + \lambda \beta} w, \paren {\alpha + \lambda \beta} \gamma}
		| c = {{Defof|Unitization of Algebra over Field}}
}}
{{eqn	| r = \tuple {u w + \gamma u + \alpha w + \lambda \paren {v w + \gamma v + \beta w}, \alpha \gamma + \lambda \beta \gamma}
}}
{{eqn	| r = \tuple {u w + \gamma u + \alpha w, \alpha \gamma} +_{A_+} \lambda \tuple {v w + \gamma v + \beta w, \beta \gamma}
}}
{{eqn	| r = \paren {\tuple {u, \alpha} \circ_{A_+} \tuple {w, \gamma} } +_{A_+} \paren {\lambda \cdot_{A_+} \paren {\tuple {v, \beta} \circ_{A_+} \tuple {w, \gamma} } }
		| c = {{Defof|Unitization of Algebra over Field}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn	| l = \tuple {u, \alpha} \circ_{A_+} \paren {\tuple {v, \beta} + \lambda \circ_{A_+} \tuple {w, \gamma} }
		| r = \tuple {u, \alpha} \circ_{A_+} \tuple {v + \lambda w, \beta + \lambda \gamma}
}}
{{eqn	| r = \tuple {u \paren {v + \lambda w} + \alpha \paren {v + \lambda w} + \paren {\beta + \lambda \gamma} u, \alpha \paren {\beta + \lambda \gamma} }
		| c = {{Defof|Unitization of Algebra over Field}}
}}
{{eqn	| r = \tuple {u v + \alpha v + \beta u + \lambda \paren {u w + \alpha w + \gamma u}, \alpha \beta + \alpha \lambda \gamma}
}}
{{eqn	| r = \tuple {u v + \alpha v + \beta u, \alpha \beta} +_{A_+} \lambda \tuple {u w + \alpha w + \gamma u, \alpha \gamma}
}}
{{eqn	| r = \paren {\tuple {u, \alpha} \circ_{A_+} \tuple {v, \beta} } +_{A_+} \paren {\lambda \cdot_{A_+} \paren {\tuple {u, \alpha} \circ_{A_+} \tuple {w, \gamma} } }
		| c = {{Defof|Unitization of Algebra over Field}}
}}
{{end-eqn}}
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:
{{begin-eqn}}
{{eqn	| l = \tuple { {\mathbf 0}_A, 1_K} \circ_{A_+} \tuple {x, \lambda}
		| r = \tuple { {\mathbf 0}_A x + 1_K x + \lambda {\mathbf 0}_A, 1_K \lambda}
		| c = {{Defof|Unitization of Algebra over Field}}
}}
{{eqn	| r = \tuple {x, \lambda}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn	| l = \tuple {x, \lambda} \circ_{A_+} \tuple { {\mathbf 0}_A, 1_K}
		| r = \tuple {x {\mathbf 0}_A + 1_K x + \lambda {\mathbf 0}_A, 1_K \lambda}
		| c = {{Defof|Unitization of Algebra over Field}}
}}
{{eqn	| r = \tuple {x, \lambda}
}}
{{end-eqn}}
So $\struct {A_+, +_{A_+}, \cdot_{A_+}, \circ_{A_+} }_K$ is a unital algebra over $K$.
{{qed}}
Category:Algebras over Fields
Category:Unitizations of Algebras over Fields
r0zdnjpoh9sc9e125fnscsgwyoc3bsv