this post was submitted on 20 Nov 2024
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No. The standard field (that is, a ring where both operations are abelian groups) on the complex numbers doesn't have a multiplicative inverse of 0; rings can't have a multiplicative inverse for the additive identity. You can create an algebra with a ring as a sub-algebra with such, but it will no longer be a ring. My preferred method is to impose such an algebra on the one-point compactification of the Complex Numbers, where the single added point is denoted as "Ω".
I started this project when I was 12, and when I could show that the results were self-consistent this was what I had settled on:
let z be a complex number that is not otherwise specified by the following equations. Note: the complex numbers contain the Real numbers, and so the following equations apply to the them as well.
0Ω=Ω0=1
z+Ω=Ω+z=zΩ=Ωz=Ω=ΩΩ
Ω-Ω=0. Ω-Ω=Ω+(-Ω)=Ω+(-1Ω)=Ω+Ω=0
The algebra described above is not associative. That is to say, (AB)C does not always equal A(BC).
Addendum: despite what my earlier statement implied, there is exactly 1 ring for which the additive identity has a multiplicative inverse: the trivial ring, which has only 1 element (that I will label as 0).
The operations are a such: 0+0=0=0*0. Note: this is also the only ring for which the additive and multiplicative identities are the same element.
I was originally going to mention it, but I didn't want to make my comment more complicated than in needed to be. Then I realized that the way I phrased it was technically inaccurate, and so this addendum exists.