Unitization of Commutative Algebra over Field is Commutative

Theorem

Let $K$ be a field.

Let $A$ be a commutative algebra over $K$.

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

Then $A_+$ is commutative.

Proof

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

Then, we have:

\(\ds \tuple {x, \lambda} \tuple {y, \mu}\)

\(=\)

\(\ds \tuple {x y + \lambda y + \mu x, \lambda \mu}\)

\(\ds \)

\(=\)

\(\ds \tuple {y x + \lambda y + \mu x, \mu \lambda}\)

$A$ is commutative, $K$ is a field

\(\ds \)

\(=\)

\(\ds \tuple {y, \mu} \tuple {x, \lambda}\)

So $A_+$ is commutative.

$\blacksquare$