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.
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.