Another definition[1] of prime in an arbitrary ring is

  p is prime if and only if p|ab implies p|a or p|b

which is equivalent to the prime ideal concept. Interestingly, these definitions mean that 0 is actually a prime in Z!

As explanation: the only number 0 divides is 0, and Z is an integral domain[2], i.e. ab = 0 implies a = 0 or b = 0, thus 0 divides at least one of a and b if 0 divides ab.

[1]: https://en.wikipedia.org/wiki/Prime_element [2]: https://en.wikipedia.org/wiki/Integral_domain