GA is a sub-theory of Peano Arithmetic (PA). If we add an induction axiom (IND) to the axioms of Ring Theory (RT) then GA is also a sub-theory of RT+IND. (We also need a weak successor axiom).
Boucher proves Lagrange's four square theorem, every number is the sum of four squares, is a theorem of GA. Since the four square theorem is not true in the integers, the integers can not be a model for GA, PA, or RT+IND.