Theories of first order of integer semigroup rings (Q1909855)

The aim of the paper is to start studying the first order theories of semigroup rings. From the author's commentary: ``In the first section we consider integer semigroup rings of free semigroups of no more than countable rank and of cyclic groups. We establish criteria of elementary equivalence of such rings. In the second section we consider integer semigroup rings of free semigroups with unit of arbitrary rank and cyclic semigroups. We describe decidable theories of such rings in the frames of the scheme-alternative hierarchy, which was defined in [\textit{Yu. M. Vazhenin}, Algebra Logika 26, No. 4, 419-434 (1987; Zbl 0647.03006)]''.