The Axiomatics of Free Group Rings

• Published in: arXiv.org 2021-12
• Main Authors: Fine, Benjamin, Gaglione, Anthony, Kreuzer, Martin, Rosenberger, Gerhard, Spellman, Dennis
• Format: Article
• Language: English
• Subjects: Commutativity; Equivalence; Group theory; Rings (mathematics); Sentences
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2112.01056

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 \({\mathbb Z}[F]\).