The theory of commutative absolutely flat rings (a.k.a. commutative von Neumann regular rings) is algebraic and furthermore it is the smallest variety containing all fields. Being a variety the category of (set-theoretic) models is arguably much better behaved than the category of fields.
How does linear algebra in modules over commutative absolutely flat rings compare to plain linear algebra, i.e. working with vector spaces? Is is it basically the same or are there some major differences?
Let's ask some more concrete questions:
Let $R$ be an commutative absolutely flat ring: Are finitely generated $R$-modules always free? Assuming the axiom of choice: Are all $R$-modules free?