You can totally divide by zero, but the ring you get when you do that is the zero ring, and it only has one element. When you start with the integers and try dividing by nonzero stuff, you can say “you can’t do that” or you can move out of the integers and into the rationals, into which the integers embed (or you can restrict yourself to only dividing by some nonzero things—that’s called localization—which is also interesting). The difference between doing that and dividing by zero is that nothing embeds into the zero ring (except the zero ring). It’s not that we can’t study it, but that we don’t want to.
Also, in the future, if you want to ask math questions, ask them on math.stackexchange.com (I’ve answered a version of this question there already, I think).
I mean if you localize a ring at zero you get the zero ring. Equivalently, the unique ring in which zero is invertible is the zero ring. (Some textbooks will tell you that you can’t localize at zero. They are haters who don’t like the zero ring for some reason.)
The theorems work out nicer if you don’t. A field should be a ring with exactly two ideals (the zero ideal and the unit deal), and the zero ring has one ideal.
We often want the field without zero to form a multiplicative group, and this isn’t the case in the ring with one element (because the empty set lacks an identity and hence isn’t a group). Indeed we could take the definition of a field to be
A ring such that the non-zero elements form a multiplicative group.
We don’t divide by zero because it’s boring.
You can totally divide by zero, but the ring you get when you do that is the zero ring, and it only has one element. When you start with the integers and try dividing by nonzero stuff, you can say “you can’t do that” or you can move out of the integers and into the rationals, into which the integers embed (or you can restrict yourself to only dividing by some nonzero things—that’s called localization—which is also interesting). The difference between doing that and dividing by zero is that nothing embeds into the zero ring (except the zero ring). It’s not that we can’t study it, but that we don’t want to.
Also, in the future, if you want to ask math questions, ask them on math.stackexchange.com (I’ve answered a version of this question there already, I think).
Thanks, I think that answers my question.
What do you mean by “you get”? Do you mean Wheel theory or what?
I mean if you localize a ring at zero you get the zero ring. Equivalently, the unique ring in which zero is invertible is the zero ring. (Some textbooks will tell you that you can’t localize at zero. They are haters who don’t like the zero ring for some reason.)
BTW, how comes the ring with one element isn’t usually considered a field?
The theorems work out nicer if you don’t. A field should be a ring with exactly two ideals (the zero ideal and the unit deal), and the zero ring has one ideal.
Ah, so it’s for exactly the same reason that 1 isn’t prime.
Yes, more or less. On nLab this phenomenon is called too simple to be simple.
We often want the field without zero to form a multiplicative group, and this isn’t the case in the ring with one element (because the empty set lacks an identity and hence isn’t a group). Indeed we could take the definition of a field to be
and this is fairly elegant.