The notion of 'negative' doesn't transfer too well to other rings. It vaguely makes sense to only look at the natural numbers (excluding 0), primality predates negative numbers as a concept after all, but from the point of view of the more general definition it is mostly arbitrary.

Primality is defined for any ring (although some rings have no primes). If you are asking about prime integers, then negative primes are themselves prime. Most of what you know of primality still holds if you add 'up to unit multiples' to all of your theroums.