3 ms·
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 multipli
by 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 agoYeah, 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.