Andrew Boucher has developed a theory called General Arithmetic (GA): http://www.andrewboucher.com/papers/ga.pdf
GA is a sub-theory of Peano Arithmetic (PA). If we add an induction schema (IND) to the axioms of Ring Theory (RT) then GA is also a sub-theory of RT+IND. (We might 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 PA or RT+IND.
I know very little about ring theory. Searching ring theory and induction I found this result: http://www.proofwiki.org/wiki/Binomial_Theorem/Ring_Theory
Are there standard results in RT that are typically proven using induction? Are there results in ring theory that can only be proven using induction?