Meadows with usual real division

Question

A "meadow" is a commutative ring with a multiplicative identity element and a total multiplicative inverse operation satisfying the two equations $(x^{-1})^{-1} = x$ and $x \times (x \times x^{-1}) = x$.

Is there a meadow in which the inverses for all reals (except zero) are the usual inverses of the field of real numbers?

References would be appreciated. Thanks.

Answer 1

Quoting from the abstract of the paper which apparently introduced meadows:

We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply that the inverse of zero is zero. All fields and products of fields can be viewed as meadows.

In other words, since $\mathbb{R}$ is a field you just need to extend it with $0^{-1} = 0$.