The axiomatics of free group rings
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 \(\mathbb{Z}[F]\).
scientific article; zbMATH DE number 7875714