The prime meadow of a meadow Let $(R,(-)^{-1})$ be a meadow, i.e. $R$ is a commutative ring and $(-)^{-1}$ is a unary operation on the underlying set of $R$ satisfying $(x^{-1})^{-1} = x$ and $x \cdot x^{-1} \cdot x = x$ for all elements $x$. (Notice that $R$ is a commutative von Neumann regular ring, and conversely every commutative von Neumann regular ring has a unique expansion to a meadow. The homomorphisms agree. So these concepts are really the same.)
Question. Let $(R,(-)^{-1})$ be a meadow. Why is $\{n_1 \cdot m_1^{-1} + \dotsc + n_k \cdot m_k^{-1} : n_i,m_i \in \mathbb{Z}\}$ a sub-meadow of $(R,(-)^{-1})$?
The set is clearly a subring, so the only thing to check is that it is closed under $(-)^{-1}$. There is an abstract reason for this, using the initial meadow and its structure derived from the universal property, but I would like to see a direct equational argument for it.
 A: The set you describe is clearly closed under addition, additive inverse, and multiplication. The issue is to have it closed under (multiplicative) inverse (division). In my view the latter closure is, though true, not entirely obvious in the initial meadow (which is described in detail by Bethke and Rodenburg in ref. 11 of the paper mentioned below, or see http://arxiv.org/pdf/0806.2256.pdf). The closure of the set you mention under inverse is obvious, however, in every meadow which expands a field with an inverse operation such as the rational numbers.
In Bergstra & Middelburg "Transformation of fractions into simple fractions in divisive meadows" (to appear in J. of Applied Logic, see also http://arxiv.org/pdf/1510.06233.pdf) it is shown (thm. 21) that from the equations of meadows it follows that each expression is (provably) equal to a sum of simple fractions. This works for open terms and in my view by being a somewhat more general result it may serve as the "explanation" based on equational reasoning which you are asking for. 
From this observation one then obtains a generalisation of what you have stated: the m_i and n_i can be taken from an arbitrary unital subring S of R. In that case the corresponding set (of sums of simple fractions with both numerator and denominator in S) is a submeadow of R as well.
For this latter observation it is required that the meadow under consideration is commutative (while your observation works in an arbitrary skew meadow just as well). In the setting of skew meadows and working with an arbitrary S instead of Z sums of product of simple fractions come into play, rather thans sums of simple fractions.
