The axiomatics of free group rings
type: Item
instance of: scholarly article
title: The axiomatics of free group rings
author: Benjamin Fine
author: Anthony Gaglione
author: Martin Kreuzer
author: Gerhard Rosenberger
author: Dennis Spellman
published in: Journal of Groups, Complexity, Cryptology
publication date: 2024-07-03T00:00:00Z
review text: In their previous articles, the authors examined 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 \(R\). Here, they study the case when \(R\) is a commutative ring with identity \(1\) not equal to \(0\). 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. The authors show that this conjecture is true in terms of axiom systems for \(\mathbb{Z}[F]\).
reviewed by: Igor Subbotin
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 7875714
zbMATH Keywords: group ring
zbMATH Keywords: elementary equivalent
zbMATH Keywords: universally equivalent
zbMATH Keywords: discriminates
zbMATH Keywords: axiomatic systems
zbMATH Keywords: quasi-identity
MaRDI profile type: Publication
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DOI: 10.46298/JGCC.2021.13.2.8796
description: scientific article; zbMATH DE number 7875714