Are there non-commutative models of arithmetic which have a prime number structure? 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?
 A: 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.
A: Yes, I believe there are.
I believe I came across such system in my research.
First you take the zero out of natural numbers and try to work with the remaining numbers
in a system where no number is left unrepresented. To do so you do a displacement: 1 represents 0, 2 represents 1 and so on.
You can check the introduction of the idea here (it is just the introduction):
"
https://repositorioaberto.uab.pt/bitstream/10400.2/1292/1/p_50_58.%20pdf.pdf
"
In such system, addition is different because 1 is the neutral element of addition:
2++1=2   
a++1=a
And generally 
a++b=a+b-1
Multiplication is therefore not commutative, because for example
2**3=3++3=5
3**2=2++2++2=(2++2)++2=3++2=4
And generally
a**b=a*b-(a-1)
But prime numbers remain the structure of this system
The new primes are simply
new prime=old prime+1
Since there is a correspondence between normal arithmetics and this system (which I have checked), the prime structure holds
There is no way to decompose a new prime other than right multiplication by two of its natural antecessor:
For example the new prime 8 can be obtained by 
7**2=8  But multiplication by two on the right is just a sort of application of the neutral element of mutiplication displaced for the new system, 
the equivalent of adding one unit: 7++2=8 
This should not therefore be considered a "legal" multiplication for the purpose of prime identification. 
Therefore primes remain primes in this non-commutative arithmetic.
I am sorry if this does not sound very rigorous. Later I can eventually provide more details and proofs. Now I am finishing a project on my daytime job.
