I'm sure this is an obvious question, but I'm having trouble finding the right words to type into Google.

I know that the definition of a ring allows that the additive identity not have a multiplicative inverse, but is this a requirement?

Specifically, is something like $\mathbb{R}\!\left[\frac{1}{0}\right]$ such that $\frac{1}{0} \cdot 0 = 1$ a ring, or does some contradiction arise from allowing the additive identity to have a multiplicative inverse?