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Qiaochu_Yuan comments on Open Thread, April 15-30, 2013 - Less Wrong
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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.