For simplicity let us assume that the
set of monomials not divisible by the leading
term of any element of **G** is finite, as in our first example
(where the monomials were
).
In this case we say the quotient ring
is finite-dimensional, and this corresponds to a
`zero-dimensional' variety (it just means
that there are only a finite set of
solutions to the polynomial system).

Then we can express any given **f** and **g**,
modulo the ideal, as say and
respectively. It follows immediately that the
representation of **f+g** is and
somewhat less immediately that the representation of
**fg** is the remainder on division
by **I** of .

This produces a * linear vector space* as is easily verified.
Let us consider the
effects of multiplying by in this vector space.

Suppose , where
the are the monomials in **A** and the are coefficients.
Suppose also that reduces (on division by I) to
, for **i = 1** to **s**.
Then it is easily seen that

That is, multiplication by acts on the space spanned by these monomials in a linear way, which we can represent by a matrix.

Obviously there's nothing special about ,
and each variable is represented by
such a matrix.
Note that since multiplication by and in the polynomial
ring * commutes*,
these matrices and must also commute.

It should also be clear that knowledge of the action of these multiplication matrices should tell us everything possible about the ideal, in some sense. So the following miracle is actually explainable.

Tue Mar 12 21:09:19 EST 1996