soft question - Slightly weakened / altered concepts of a field

The meadow (as defined in the question, and in the paper linked) is an "equational theory".

A meadow is a set $A$ together with operations $0,1,+,-,cdot,{}^{-1}$ such that $(A,0,1,+,-,cdot)$ is a commutative ring with unit, and identities
$$
(x^{-1})^{-1} = x
\
xcdot(xcdot x^{-1}) = x
$$
hold for all $x in A$.

As with all equational theories, this tells us what are the morphisms, subalgebras, ideals, products, quotients, and so on. So: if $A, B$ are meadows, then a map $f : A to B$ is a homomorphism iff
$$
f(0)=0\
f(1)=1\
f(x+y)=f(x)+f(y)
\
f(-x)=-f(x)\
f(xy) = f(x)f(y)
\
f(x^{-1})=f(x)^{-1}
$$
(Some of these follow from the others, but abstractly you just say it preserves all the operations.)

A submeadow of a meadow $A$ is a subset $B subseteq A$ such that
if $x,y in B$, then
$$
0, 1, x+y, -x, xcdot y, x^{-1} in B
$$

An important example of a meadow is a field, with the usual partial operation $x^{-1}$ enhanced to a total operation by defining $0^{-1}=0$. This is called a zero totalized field.

More interesting examples are products of fields. For example $mathbb{Md}_6 = mathbb F_2 times mathbb F_3$.
Theorem: Any meadow is (up to isomorphism) a submeadow of a product of zero totalized fields.

As Robin Chapman noted (quoted in the paper mentioned): Take a meadow and forget the inverse operation, and you have a von Neumann regular ring; start with a von Neumann regular ring with unit, there is a unique way to define the inverse making it a meadow.