3 ms·

I did once try to run through all the definitions and their generalisations, and it basically boils down to the fact that the zero ring is not a field. This giv

I did once try to run through all the definitions and their generalisations, and it basically boils down to the fact that the zero ring is not a field. This gives the following stacktrace as for why you can't make 1 a prime: - A prime number is a positive prime element of Z - A prime element is an element p which is not a unit such that whenever p divides ab then p divides a or p divides b (or both) - An element which generates a prime ideal is a prime element - A ring divided by a prime ideal is an integral domain - All integral domains have a field of fractions - The zero ring is not a field Make 1 a prime at the first step and things go wrong at the final step. Most of these definitions have a good argument, except the first arbitrarily excludes the non-positive numbers. This includes 0 which is a perfectly acceptable prime according to all other definitions (and unlike the negative primes, is not simply the negative of a prime number).

tgv 3y ago
The exclusion of negative numbers is not arbitrary. Depending on your exact definition, you would be left with the current prime numbers, these plus their negatives, or none at all (since 2 is divisible by -2).
- contravariant 3y ago
  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.
- gizmo686 3y ago
  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.
kevinventullo 3y ago
Why stop there; why can’t we consider the zero ring a field? It has addition, subtraction, multiplication, and (vacuously!) every nonzero element has a multiplicative inverse. I realize the definition of fields usually includes the axiom 0 != 1, but I think this is just as arbitrary as “1 is not prime”. You can even extend this to scheme theory, where Spec of the zero ring is the empty set, and conversely the zero ring is the ring of regular functions on the empty set. Note that the empty set is always open in the Zariski topology! Anyway I’m not actually seriously advocating for this position, just that it does seem to be consistent.
- contravariant 3y ago
  Yeah, best argument I can come up with is that the multiplication over the zero ring minus 0 does not form a group. But you'd need to modify the usual definition of a field for this to look natural. There is quite a lot of research on a zero field, which shows up sometimes (never as a conceret object, but more like A_p is to the field of p elements what A_0 is to the zero field). Apparently you can't just use the zero ring, though the reasons why are beyond my pay-grade.

tgv 3y ago

The exclusion of negative numbers is not arbitrary. Depending on your exact definition, you would be left with the current prime numbers, these plus their negatives, or none at all (since 2 is divisible by -2).

contravariant 3y ago
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.
gizmo686 3y ago
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.

kevinventullo 3y ago

Why stop there; why can’t we consider the zero ring a field? It has addition, subtraction, multiplication, and (vacuously!) every nonzero element has a multiplicative inverse. I realize the definition of fields usually includes the axiom 0 != 1, but I think this is just as arbitrary as “1 is not prime”. You can even extend this to scheme theory, where Spec of the zero ring is the empty set, and conversely the zero ring is the ring of regular functions on the empty set. Note that the empty set is always open in the Zariski topology! Anyway I’m not actually seriously advocating for this position, just that it does seem to be consistent.

contravariant 3y ago
Yeah, best argument I can come up with is that the multiplication over the zero ring minus 0 does not form a group. But you'd need to modify the usual definition of a field for this to look natural. There is quite a lot of research on a zero field, which shows up sometimes (never as a conceret object, but more like A_p is to the field of p elements what A_0 is to the zero field). Apparently you can't just use the zero ring, though the reasons why are beyond my pay-grade.