[FOM] Boolean Rings : Commutative Rings :: Propositional Logic : A 'free' Logic?

[Rex Butler]

Boolean algebras are associated to Propositional logic.  Various
algebraic generalizations have associations with other logics.  For
example, in the generalization to Heyting algebras is found the
association with Intuitionistic Logic.  Likewise, other generalized
classes of lattices are associated with other logics.  Now, by
analogy, what if Boolean rings (with the appropriate derived
operations) are used instead of Boolean algebras and the same process
of generalization is considered?   The obvious candidate for
generalization here is the removal of the identity x^2 = x, whereby
the class of Boolean rings is widened to the class of commutative
rings at large.

Which brings me to my question.  Thinking of commutative rings as
boolean rings without the identity x^2 = x, to what 'free' logical
systems might commutative rings be associated with?

Regardless of the answer, it seems like commutative algebra and
algebraic geometry have a protological flavor unique among other
mathematical disciplines. This might suggest the primacy of algebraic
geometry, where (and I state this loosely) 'boolean algebraic
geometry' occurs early in the history of what would become the subject
as a whole.


Rex Butler