Elementary and universal equivalence of group rings. Issue 7 (2nd July 2020) - Fine, Benjamin; Gaglione, Anthony M.; Rosenberger, Gerhard; Spellman, Dennis
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 T 0 and a ring as a model of T 1 . Moreover, we view the class of group rings as a subclass of the model class of T 2 . The paper consists of two parts. In Part (I), we prove 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 . In Part(II) we let F be a rank 2 free group and Z be the ring of integers. We show that if G is universally equivalent to F with respect to L 0 and R is universally equivalent to Z with respect to L 1, then, R [ G ] is universally equivalent to Z [ F ] with respect to L 1 . Furthermore, we show that, if R is universally equivalent to Z with respect to L 1 and R [ G ] is universally equivalent to Z [ F ] with respect to L 1, then, G is universally equivalent to F with respect to L 0 .