Regular representations of finite-dimensional separable semisimple algebras and Gröbner bases

Let be a semisimplen-dimensional commutative algebra over a field . It is easy to see that, given a basis of , the transposes of the matrices over that represent regularly with respect to can be simultaneously diagonalized over many fields. Using the multiplication table of the algebra we construct an ideal of given in terms of a Gröbner basis of the ideal with respect to a total degree lexicographic monomial ordering and show that is isomorphic to . We will then use Gröbner basis properties to prove the properties of the algebra.