The Axiomatics of Free Group Rings

Groups Complexity Cryptology

article

Benjamin Fine¹  Anthony Gaglione²  Martin Kreuzer³  Gerhard Rosenberger⁴  Dennis Spellman⁵ 

[1] Department of Mathematics, Fairfield University;Department of Mathematics;Faculty of Informatik und Mathematik, University Passau;Fachbereich Mathematik, University of Hamburg;Department of Mathematics, Temple University

关键词: Group ring;    elementary equivalent;    universally equivalent;    discriminates;    axiomaticsystems;    quasi-identity.;   

DOI  :  10.46298/jgcc.2021.13.2.8796

来源: Episciences

【 摘 要 】

In [FGRS1, FGRS2] the relationship between the universal and elementarytheory of a group ring R[G] and the corresponding universal and elementary theory of theassociated group G and ring R was examined. Here we assume that R is a commutativering with identity 1 0. Of course, these are relative to an appropriate logical languageL0, L1, L2 for groups, rings and group rings respectively. Axiom systems for these wereprovided in [FGRS1]. In [FGRS1] it was proved that if R[G] is elementarily equivalent toS[H] with respect to L2, then simultaneously the group G is elementarily equivalent tothe group H with respect to L0, and the ring R is elementarily equivalent to the ring Swith respect to L1. 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 madethat the universal sentences true in Z[F] are precisely the universal sentences true in Fmodified appropriately for group ring theory and the converse that the universal sentencestrue 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].

【 授权许可】

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