## Benjamin Fine ; Anthony Gaglione ; Martin Kreuzer ; Gerhard Rosenberger ; Dennis Spellman - The Axiomatics of Free Group Rings

gcc:8796 - journal of Groups, complexity, cryptology, December 6, 2021, Volume 13, Issue 2 - https://doi.org/10.46298/jgcc.2021.13.2.8796
The Axiomatics of Free Group Rings

Authors: Benjamin Fine ; Anthony Gaglione ; Martin Kreuzer ; Gerhard Rosenberger ; Dennis Spellman

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 ${\mathbb Z}[F]$.

Volume: Volume 13, Issue 2
Published on: December 6, 2021
Accepted on: December 3, 2021
Submitted on: December 3, 2021
Keywords: Mathematics - Group Theory,Mathematics - Logic,Mathematics - Rings and Algebras