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):
In such system, addition is different because 1 is the neutral element of addition:
Multiplication is therefore not commutative, because for example
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.