you may be referring to an algebra over a field, which has a dual structure as a ring and a vector space, which share the same additive operation. for compatability reasons, the multiplication is required to be bilinear (so it satisfies the distributive laws). the usual examples are associative and unital (the ring has unity), but other examples do exist.
some typical examples of such algebras are: nxn matrices over a field, polynomials over a field (F[x]), and the quaternions. for any vector space V, the set End(V) = Hom(V,V) is another example (in which the multiplication is function composition).
as a vector space, we can consider such things as dim(A), bases, and scalar multiples. as a ring we can talk about ideals, and the group of units (the group of units of the algebra of nxn matrices is the general linear group of degree n over F).
i'm not sure what you are trying to ask with the specific relation in your question.