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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?

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The four square theorem is not true in the integers? Well, it's true in the non-negative integers, and that's what PA is about. –  Gerry Myerson Nov 29 '12 at 3:38
    
Yes. I was asking about ring theory. Aren't the integers a model of ring theory? –  Russell Easterly Nov 29 '12 at 3:49
    
Any property in a ring which involves natural numbers might rely on induction. For example, that $m \cdot 1 $ times $n \cdot 1$ is $mn \cdot 1$. Some standard results in RT using induction, I believe are that polynomials over fields have division algorithms, and that a type of ring in which every non-divisible element is expressible as a product of prime-like elements, are uniquely expressed as prime-like elements, analogous to the fundamental theorem of algebra. (These rings are called unique factorization domains). –  insert pretentious shit here Jul 19 '13 at 20:57
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