View source for Finite-dimensional associative algebra

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An associative algebra (cf. [[Associative rings and algebras|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|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|Wedderburn–Artin theorem]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German) {{MR|1541390}} {{ZBL|1032.00002}} {{ZBL|1032.00001}} {{ZBL|0903.01009}} {{ZBL|0781.12003}} {{ZBL|0781.12002}} {{ZBL|0724.12002}} {{ZBL|0724.12001}} {{ZBL|0569.01001}} {{ZBL|0534.01001}} {{ZBL|0997.00502}} {{ZBL|0997.00501}} {{ZBL|0316.22001}} {{ZBL|0297.01014}} {{ZBL|0221.12001}} {{ZBL|0192.33002}} {{ZBL|0137.25403}} {{ZBL|0136.24505}} {{ZBL|0087.25903}} {{ZBL|0192.33001}} {{ZBL|0067.00502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) {{MR|0000595}} {{ZBL|0023.19901}} {{ZBL|65.0094.02}} </TD></TR></table>

====Comments====
Skew-fields are also known as division algebras, cf. [[Division algebra|Division algebra]].

The representation theory of finite-dimensional (associative) algebras is a very active branch of mathematics nowadays (1988). Cf., e.g., [[#References|[a1]]]–[[#References|[a2]]] and [[Quiver|Quiver]] and [[Representation of an associative algebra|Representation of an associative algebra]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Pierce, "Associative algebras" , Springer (1980) {{MR|0890331}} {{MR|0674652}} {{ZBL|0671.16001}} {{ZBL|0497.16001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.M. Ringel, "Tame algebras and integral quadratic forms" , ''Lect. notes in math.'' , '''1099''' , Springer (1984) {{MR|0774589}} {{ZBL|0546.16013}} </TD></TR></table>

[[Category:Associative rings and algebras]]