Abstract
In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was 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 the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${\mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${\mathbb Z}[F]$.
Highlights
In [FGRS1, FGRS2] the relationship was examined between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R
In [FGRS1] it was proved that if R[G] is elementarily equivalent to S[H] with respect to L2, simultaneously the group G is elementarily equivalent to the group H with respect to L0, and the ring R is elementarily equivalent to the ring S with respect to L1
Examining the universal theory of the free group ring Z[F ] the hazy conjecture was made that the universal sentences true in Z[F ] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in Z[F ] modified appropriately for group theory
Summary
In [FGRS1, FGRS2] the relationship was examined between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R. Myasnikov and Remeslennikov [MR] have given axiom systems for the universal theory of non-abelian free groups In particular they proved that if F is a non-abelian free group the universal theory of F is axiomatized by (see section 2 for relevant definition) the diagram of F , the strict universal Horn sentences of L0[F ] true in F and group commutative transitivity (see sections 3 and 4 for relevant definitons). In this paper we extend this to axiom systems for free group rings and prove that the universal theory of a free group ring Z[F ] is axiomatized by the diagram of Z[F ], the strict universal Horn sentences of L2[Z[F ]] true in Z[F ] and ring commutative transitivity when the models are restricted to group rings.
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