When I first read this question it was grey, but it's a good question, perhaps too technical for the most readers.

There are a few similar concepts

* irreducible elements: https://en.wikipedia.org/wiki/Irreducible_element

> In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

* prime element: https://en.wikipedia.org/wiki/Prime_element

> An element p of a commutative ring R is said to be prime if it is not zero or a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b.

In the integers the definitions are equivalent, but in more general cases they are different.

* prime ideal: https://en.wikipedia.org/wiki/Prime_ideal

> An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, [and] P is not equal to R for the whole ring.

The prime ideals are a "generalization" of the prime numbers to more general sets, that have elements that may not even look like numbers. It's a very important idea in algebra, and it's not an straightforward generalization because the elements in the new set may not have all the properties that the number have. (More details in the links to Wikipedia an in any advance algebra book.)

In particular in the integers if p is a prime then the set pZ={..., -3p, -2p, -p, 0, p, 2p, 3p, ...} is a prime ideal.

Also in the integers, if p !=0 and pZ is a prime ideal, then p is a prime.

Therefore, in the integers there is an almost 1 to 1 relation between prime elements and prime ideals. So

1) This is a "good" generalization

2) The number 0 is a nasty exception

So the question in the previous comment is "Why is it a good idea to make a nasty exception for 0? Would it be better to redefine the primes numbers to include 0?"

I personally would prefer a redefinition of the prime number to include 0, but it is not the standard definition :( .

tl;dr: Please don't downvote the previous comment unless you know what is a "prime ideal".

If you consider the fundamental theorem of arithmetic, it makes the work much easier work if you don't include 0 in prime numbers (the same justification holds for 1).

If you consider "prime" to mean "A number is evenly divisible only by itself and 1" then 0 is not prime, as 0 is not divisible by itself.

It definitely looks a bit prime.

If it divides into a product, it divides into one of the factors.

The character of an integral domain can be zero.

If zero isn't prime, then it has no factorisation.