PTFA group rings embed in skew fields of fractions

Fix a finite series $1=G_0⊴G_1⊴\cdots ⊴G_m=G$ as in [F1]. Its base ring $\mathbb{Q}[G_0]=\mathbb{Q}$ has no zero divisors. For each $j<m$, apply [F3] to $G_j⊴G_{j+1}$ and its torsion-free abelian quotient. Induction gives absence of zero divisors in $\mathbb{Q}[G]$. The domain supplier keeps the conjugation automorphisms and cocycle units of the extension in its leading-coefficient product, so this induction assumes neither a splitting nor normality of each $G_j$ in the final group.

By [F4], the same finite subnormal series gives finite left Folner sets in $G$. That supplier constructs finite sets directly across each group extension, with only finitely many coset lifts for each test, rather than assuming an invariant mean or adding a choice principle.

Apply [F5] to the ring from step 1.1 and the finite sets from step 1.2. It gives nonzero common right multiples $ax=by$ of every nonzero $a,b$, and the group-inversion anti-automorphism gives nonzero common left multiples. It then constructs a division ring from equivalence classes of right-linear maps on nonzero right ideals, embeds $\mathbb{Q}[G]$ injectively, and shows every element of that division ring is $a{s}^{-1}$ with $0\mathrm{\ne }s\in \mathbb{Q}[G]$. These are exactly both Ore conditions and the required skew field of fractions. The length-zero case is already included by the base field.