Injective minimal modules (Q1842384)

A complete theory \(T\) is called strongly minimal if it has infinite models and every first order definable subset of a model of \(T\) is finite or cofinite. A structure \(M\) is called strongly minimal if the complete theory of \(M\) is strongly minimal. Strongly minimal structures are among the basic building blocks of more complicated structures. This paper continues earlier work of the author [Abelian Groups and Modules, Tomsk 65-79 (1984; Zbl 0658.20030)] and with \textit{G. Punninski} [Algebra Logika 30, No. 5, 557-567 (1991; Zbl 0774.03021)] on the problem of obtaining purely algebraic characterizations of strongly minimal modules over a ring \(R\). The problem reduces to that of faithful minimal modules; in this case \(R\) is a domain embeddable in a division ring. If \(R\) is a left Ore domain, a minimal faithful right module \(M\) over \(R\) is precisely a divisible module with \(\{m \in M : mr=0\}\) finite for all \(0 \neq r \in R\). For a ring \(R\) with infinite centre \(C\), the author considers the ring of first-order definable \(C\)-endomorphisms \(T=T(M)\); \(R \subset T(M) \subset \text{End}(M_C)\). The key result is that for infinite faithful \(M\), \(M_R\) is minimal iff \(M_T\) is minimal, in which case \(T\) is a right noetherian Ore domain. Since minimal modules are divisible, it is natural to ask which injective modules are minimal. The author first shows that \(\aleph_0\)-injective and \(RD\)-injective faithful minimal modules are injective. A complete characterization of injective minimal modules over commutative rings is given.