General Algebra and Linear Algebra

Dear All,

I have come across an interesting property of some special
equations on algebras that seem to connect to congruence relations. The
following is an attempt to formulate the idea of the
equations in a more general form. I would be most interested in
any small pointers you might have to the literature. Please excuse any
misuses of language!

Let t_i be a set of terms on the variables x_j in
an algebra with set A.
Let a_i be some real numbers.
Let v map A to the reals R such that the equation
sum_i a_i v(t_i) = 0
is satisfied for all values of the x_j selected from A. It is clear that
the set of v satisfying such equations forms a vector space that is
a subspace of R^A.

It looks like it should be called something like "solutions of
systems of equations over valuations of an algebra"
but of course "valuations" already has a well defined meaning.
Perhaps it has some more meaning when we generalise R to allow
it to be an abelian group and the a_i to be integers: then an
equation like
v(x*y) - v(x) - v(y) = 0
corresponds to v being a homomorphism of A into the group R.

I hope for some clever suggestions,

Best,

Tim