A prime element of a ring (like the integers) is defined to be a non-unit p that satisfies p|ab => p|a or p|b. A unit is defined to be a number that you can div

A prime element of a ring (like the integers) is defined to be a non-unit p that satisfies p|ab => p|a or p|b. A unit is defined to be a number that you can divide by. Therefore the units in the integers are {-1,+1}. Based on that, -2 is prime element and -1 is not. The statement of unique factorisation is more complicated when using this definition of prime, but it allows us to talk about primes in the Gaussian and Eisenstein integers, as well as other Unique Factorisation Domains.