Hello everyone, was wondering if I could get some help with this.
I understand that if we have a field K with subfield F, we can think of K as a vector space over F. But if K is just an integral domain and not a field, can we still think of it as a vector space over the field F?
yes. any ring R can be made into a vector space over a subfield F, provided that F lies in the center of R (which will always happen if R is commutative). this is because (R,+) is an abelian group, and distributivity of multiplication over additon, and the associativity of multiplication ensure that the ring multiplication will satisy the axioms for scalar multiplication (the requirement that F lie in the center of R is to ensure that the multiplication in R is bi-linear, so we need r(as) = a(rs), where a is in F, and r,s are in R).
there is, in fact, an name for this situation, such a ring is called an (associative) algebra. examples are: F[x], Mn(F) (nxn matrices over F), the continuous real-valued functions defined on a subset A of R, the quaternions (over the reals, but not over the complex field), and the set of linear operators on a Hilbert space.