Is something missing here? As I understand it, that general definition works like this:
6 divides 49, but does not divide either 4 or 9. Therefore, 6 is not prime.
But I'm pretty sure 1 cannot divide the product of two integers m, n* while failing to divide m and n individually. So while that general definition does exclude composite numbers, it doesn't seem to exclude 1?
The correct "general" definition also requires the prime element to be non-invertible and non-zero, thus 1 is excluded.
It should be noted that this definition excludes "composite numbers" only when the ring is an integral domain. In a general commutative ring this isn't necessarily true.
6 divides 49, but does not divide either 4 or 9. Therefore, 6 is not prime.
But I'm pretty sure 1 cannot divide the product of two integers m, n* while failing to divide m and n individually. So while that general definition does exclude composite numbers, it doesn't seem to exclude 1?