Meadows with usual real division 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.

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 $0^{-1} = 0$.