Peano Arithmetic (PA) models have a prime number structure and commutativity of addition and multiplication. Presburger arithmetic (PrA) models of arithmetic have addition without multiplication and without a prime number structure. However, as I understand it, when we add the axiom of induction and the multiplication relation to PrA, we get a model of the multiplicative natural numbers from which commutativity follows, as in PA.
My question is whether induction plus multiplication implies a prime structure with commutativity, or if there is a model of arithmetic which has a prime number structure and is not commutative? Does the definition of "prime" rely on the commutativity of multiplication of natural numbers?
Best Answer
Commutativity is not necessary for the notion of primes. For instance, consider the Hurwitz integers, namely quaternions whose components are either all integers or all half-integers:
$$ H = \{ a + bi + cj + dk : (a, b, c, d) \in \mathbb{Z}^4 \cup (\mathbb{Z} + \frac{1}{2})^4 \} $$
These form a non-commutative ring. Moreover, there is a version of unique factorisation discussed here:
http://www.m-hikari.com/imf/imf-2012/41-44-2012/perngIMF41-44-2012.pdf
A factorisation of a non-zero Hurwitz integer into irreducibles is unique up to applying operations known as unit-migration, recombination, and meta-commutation.