# Finite-dimensional associative algebra

An associative algebra (cf. Associative rings and algebras) $A$ that is also a finite-dimensional vector space over a field $F$ such that

$$\alpha ( ab) = \ ( \alpha a) b = \ a ( \alpha b)$$

for all $a, b \in A$, $\alpha \in F$. The dimension $n = \mathop{\rm dim} _ {F} A$ of the space $A$ over $F$ is called the dimension of the algebra $A$ over $F$. It is also customary to say that the algebra $A$ is $n$- dimensional. Every $n$- dimensional associative algebra $A$ over a field $F$ has a faithful representation by matrices of order $n + 1$ over $F$, that is, there is an isomorphism of the algebra $A$ onto a subalgebra of the algebra of all square $( n + 1)$- matrices over $F$. If $A$ has an identity, then it has a faithful representation by matrices of order $n$ over $F$.

Let $e _ {1} \dots e _ {n}$ be a basis of the vector space $A$ over $F$( it is also called a basis of the algebra $A$), and suppose that

$$e _ {i} e _ {j} = \ \sum _ {k = 1 } ^ { n } \gamma _ {ij} ^ {k} e _ {k} ,\ \ \gamma _ {ij} ^ {k} \in F.$$

The elements $\gamma _ {ij} ^ {k}$ of $F$ are called the structure constants of the algebra $A$ in the given basis. They form a tensor of rank three in the space $A$.

### Main theorems concerning finite-dimensional associative algebras.

The Jacobson radical of a finite-dimensional associative algebra is nilpotent and, if the ground field is separable, it splits off as a semi-direct summand (see Wedderburn–Mal'tsev theorem). A semi-simple finite-dimensional associative algebra over a field splits into a direct sum of matrix algebras over skew-fields. If the ground field $F$ is algebraically closed, then a semi-simple finite-dimensional associative algebra splits into a direct sum of full matrix algebras over $F$. The simple finite-dimensional algebras are just the full matrix algebras over skew-fields (Wedderburn's theorem). In particular, a finite-dimensional associative algebra without zero divisors is a skew-field. The following are the only finite-dimensional associative algebras with division (that is, skew-fields) over the real field: the real field, the complex field and the skew-field of quaternions (Frobenius' theorem).

Many of the structural properties of finite-dimensional associative algebras mentioned here also hold in the larger classes of Noetherian and Artinian rings (see, e.g., Wedderburn–Artin theorem).

#### References

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