Elementary and universal equivalence of group rings. Issue 7 (2nd July 2020)

• Record Type: Journal Article
• Authors: Fine, Benjamin; Gaglione, Anthony M.; Rosenberger, Gerhard; Spellman, Dennis
• Is Part Of: Communications in algebra. Volume 48:Issue 7(2020)
• Page Start: 2740
• Page End: 2749
• Publication Date: 2020-07-02
• Subjects: Group ring -- elementary theory -- universal theory -- quasivariety -- model theory; Primary 03065 -- 16534 Secondary 20C05 -- 20C07; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2019.1710158
• Languages: English
• ISSNs: 0092-7872

Abstract

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 .