The Axiomatics of Free Group Rings

In [FGRS1,FGRS2] the relationship between the universal and elementary theoryof a group ring $R[G]$ and the corresponding universal and elementary theory ofthe associated group $G$ and ring $R$ was examined. Here we assume that $R$ isa commutative ring with identity $1 \ne 0$. Of course, these are relative to anappropriate logical language $L_0,L_1,L_2$ for groups, rings and group ringsrespectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] itwas proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to$L_{2}$, then simultaneously the group $G$ is elementarily equivalent to thegroup $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalentto the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ freegroup and $\mathbb{Z}$ be the ring of integers. Examining the universal theoryof the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that theuniversal sentences true in ${\mathbb Z}[F]$ are precisely the universalsentences true in $F$ modified appropriately for group ring theory and theconverse that the universal sentences true in $F$ are the universal sentencestrue in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paperwe show this conjecture to be true in terms of axiom systems for ${\mathbbZ}[F]$.Comment: final journal versio