Elementary and universal equivalence of group rings

Abstract

We introduce three first-order languages with equality Ln, n = 0, 1, 2. We list axioms Tn expressed in Ln and view a group as a model of and a ring as a model of T1. Moreover, we view the class of group rings as a subclass of the model class of T2. The paper consists of two parts. In Part (I), we prove that if ] is elementarily equivalent to with respect to L2, then, simultaneously the group G is elementarily equivalent to the group H with respect to L0 and the ring R is elementarily equivalent to the ring S with respect to In Part(II) we let F be a rank 2 free group and be the ring of integers. We show that if G is universally equivalent to F with respect to L0 and R is universally equivalent to with respect to L1, then, is universally equivalent to with respect to L1. Furthermore, we show that, if R is universally equivalent to with respect to L1 and is universally equivalent to with respect to L1, then, G is universally equivalent to F with respect to L0.