That may be true, but I’m with ultrafilter on that. I’d like to see a proof of it.
The real definition? 
Not all rings are unique factorization domains. Not even all Euclidean domains are.
The real definition generalizes the property that if p | ab, then p | a or p | b. In general, what one speaks of is a prime ideal in a ring. Given ideals A and B with the product ideal AB contained in P, either A is contained in P or B is. Restated more clearly, the quotient ring R/P is an integral domain – it has no zero-divisors. A prime element p is one such that the ideal it generates, (p), is prime. Since all ideals in Z are primitive (can be generated by a single element), there are no prime ideals other than those generated by primes.
As an example of an integral domain which is not a UFD, consider Z[sqrt(-5)]. The element 6 = 2*3, where 2 and 3 are prime. Also 6 = (1+sqrt(-5))(1-sqrt(-5)), and those factors are also prime.
How does that relate to regarding 1 as a kinda-sorta prime?
But that would include 1!
Yahbut, all that does not exclude the possibility of 1 as a prime, or as John says, a perfect number according to that formula.
Z/(1) is not an integral domain.
I wonder how many people read The Straight Dope in print (e.g., in the Chicago Reader), but never got round to finding the SDMB and don’t know what a ‘Doper’ is?
I’m not arguing that 1 is considered a prime. I’m just saying that John’s comment about quasi-prime is OK. The definition of integral domains has nothing to do with it, other than the definition has been worked out just so 1 is not a prime under that generalization.
My locals don’t carry the column anymore, sadly, but does the column print the staff reports?
The “real definition” specifically excludes units (i.e., ring elements that have a multiplicative inverse) as primes.
Heck, even the integers don’t have unique factorization, if you include the negatives. 7 isn’t just 17, it’s also (-1)(-7). So in such domains, we can’t define primes as “numbers whose only divisors are 1 and the number itself” (at least, we can’t if we want there to be any primes at all). Fortunately, there’s another definition of primes which generalizes in a non-empty way to such domains, and which agrees with the grade-school definition for the positive integers. So that’s the better definition to use, and under that definition, 1 isn’t a prime.
Prime Ideals and Integral Domains are defined that way (keeping the ideal generated by 1 from being prime) because 1 is not a prime, not the other way around. Some don’t define them that way–Mathworld seems to indicate that some authors allow non-proper prime ideals. In order to restrict prime ideals to proper ideals, you either have to say that the zero ring is not commutative, as they mention on that Mathworld page (although, it’s hard to see how a single ring element would not commute with itself), or you limit the definition of Integral Domain to structures that have an identity not equal to zero, as Hungerford does (in Algebra) but Mathworld doesn’t (but, again, the zero ring satisfies all the other constraints–except that one). Hungerford says that the reason for setting things up that way is because of historical and technical reasons. The historical reason seems to be that 1 is not considered a prime. When I asked for a proof earlier, I was interested in the technical reasons maybe.
Well, the proper definition of “unique factorization” is “up to rearrangement of the factors and units”. That is, since -1 is a unit in Z, -1*-7 counts as the same as 7.