Associative Algebra

Definition 1.1. Let $𝔽$ be a field. A set $\mathcal{𝒜}$ endowed with the three operations

$$\times :\mathcal{𝒜}\times \mathcal{𝒜}\to \mathcal{𝒜}$$

$$\cdot :𝔽\times \mathcal{𝒜}\to \mathcal{𝒜}$$

$$+:\mathcal{𝒜}\times \mathcal{𝒜}\to \mathcal{𝒜}$$

is an associative algebra over $𝔽$ with unit $1\in \mathcal{𝒜}$ if the following conditions hold for all $a,b,c\in \mathcal{𝒜}$ and $\alpha ,\beta \in 𝔽$.

(i)

$a+b=b+a$

(ii)

$(a+b)+c=a+(b+c)$

(iii)

$\exists <!--\; FUNCTION\; APPLICATION\; -->0\in \mathcal{𝒜}$ such that $0+a=a$

(iv)

$\exists <!--\; FUNCTION\; APPLICATION\; -->-a\in \mathcal{𝒜}$ such that $(-a)+a=0$

(v)

$\alpha \cdot (\beta \cdot a)=(\alpha \cdot \beta )\cdot a$

(vi)

$(\alpha +\beta )\cdot a=\alpha \cdot a+\beta \cdot a$

(vii)

$\alpha \cdot (a+b)=\alpha \cdot a+\alpha \cdot b$

(viii)

$1\cdot a=a$ where $1$ is the unit of $𝔽$

(ix)

$\alpha \cdot (a\times b)=(\alpha \cdot a)\times b=a\times (\alpha \cdot b)$

(x)

$a\times (b\times c)=(a\times b)\times c$

(xi)

$1\times a=a\times 1=a$ where $1$ is the unit of $\mathcal{𝒜}$

(xii)

$a\times (b+c)=a\times b+a\times c$

(xiii)

$(a+b)\times c=a\times c+b\times c$

Example 1.2. The set of all polynomials in one variable is an commutative associative algebra.

Example 1.3. The set $\mathit{Ma}{t}_{n\times n}\left(ℂ\right)$ ($M_{n\times n}\left(ℂ\right)$), which is the set of all $n\times n$ matrix with entries in $ℂ$, is a noncommutative associative algebra.

Example 1.4. Given a vector space $\mathcal{𝒱}$ over $𝔽$, the set $\mathit{End}\left(\mathcal{𝒱}\right)$, which is the set of all endomorphism of $\mathcal{𝒱}$, is an algebra. Note that, an endomorphism of a vector space is a linear transformation.

Note 1.5. Most of the time, we abbreviate an associate algebra simply as an algebra.