Yes, exactly. The name "Algebra" for this structure, by the way, comes from the fact that the set of functions, in x, say, over a field, form a vector space with ordinary addition and "scalar multiplication" but also have a natural product: f*g(x)= f(x)*g(x). (Here, f(x) and g(x) are members of the field so the multiplication on the right is the field multiplication.)