Minimal modules over Prüfer rings (Q1803025)

An algebraic system \(A\) is called minimal if any formula with parameters in \(A\) defines in it a finite or cofinite subset. Minimal algebraic systems have been studied both in the general case [\textit{B. I. Zil'ber} and \textit{V. P. Smurov}, ``On minimal structures'', in: Abstr. 9th All- Union Conf. Math. Logic, Leningrad (1988), p. 65] and for specific signatures [\textit{K.-P. Podewski}, Math.-Phys. Semesterber., n. F. 22, 193-197 (1975; Zbl 0316.16001)]. In the present paper we continue the investigation, begun by \textit{V. A. Kuzicheva}, ``Minimal modules'', in: Abelian Groups and Modules, 65-79 (Tomsk, 1984)], of minimal modules over a ring, and we are mainly interested in commutative rings. In Section 1 we study minimal modules over an arbitrary ring. We prove, in particular, that the class of all rings having minimal modules coincides with the class of rings admitting a nontrivial homomorphism into a division ring. We obtain some results on decomposing minimal modules over an arbitrary ring. In Section 2 we study minimal modules over commutative rings. We obtain a purely algebraic characterization of such modules. We prove that if \(M\) is a faithful indecomposable minimal module over a commutative ring \(R\), then either it is a vector space over the field of quotients of \(R\) or \(| R|\leq 2^ \omega\). In the latter case, the theory \(T(M)\) is \((2^ \omega)^ +\)-categorical. We characterize commutative rings \(R\) admitting minimal modules with \(R\)-torsion. In Section 3 we define the results of Section 2 in the case of commutative Prüfer rings. In particular, we completely describe the minimal modules over such rings and prove that all faithful minimal modules are injective in this case.