# Do we have negative prime numbers?

Do we have negative prime numbers?

$..., -7, -5, -3, -2, ...$

When we first encounter prime numbers, we do so in the context of the positive integers. The significant point about this context is that $$+1+1$$ is the only unit (the only positive integer with a multiplicative inverse). So the question here does not really arise. Often, when the main focus of work is the positive integers, the word prime will be used to imply a positive integer.
As soon as we start to extend this to the integers, and in particular, to consider the integers as having the structure of a ring, we add in a second unit $$−1-1$$ with $$(−1)2=1(-1)^2=1$$. Even in this context it is possible to define the prime numbers as positive integers without too much inconvenience.
But if we extend further and add $$ii$$ with $$i2=−1i^2=-1$$ as another unit – note that $$i⋅−i=1i\cdot -i=1$$, we are in a different world. For example, $$2=(1+i)(1−i)2=(1+i)(1-i)$$ and $$(1+i)=i(1−i)(1+i)=i(1-i)$$ so that $$2=i(1−i)22=i(1-i)^2$$. Are these factorisations of $$22$$ to be taken as the same or different?
So very soon, in the context of ring theory and the theory of algebraic integers, we start talking about prime ideals (initially thought of as all the multiples of some prime $$pp$$ – but extended beyond that idea too – an ideal which consists of all the multiples of a single element is called principal). And it is somewhat natural, if the ideal is principal, to call the generator a prime element of the ring. However, the primes are then only identified up to multiplication by units – $$1+i1+i$$ generates the same ideal as $$1−i1-i$$. One of the reasons for using ideals is that the uniqueness of factorisation can be maintained in this larger context. In $$Z\mathbb Z$$ both $$22$$ and $$−2-2$$ generate the same ideal.