Submission date: 2 December 2021

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 ≠ 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 ℤ be the ring of integers. 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. In this paper we show this conjecture to be true in terms of axiom systems for ℤ[F].

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Full text arXiv 2112.01056: pdf, ps.