Strongly minimal modules
The paper is a detailed report of what is known about strong minimality in (right) modules over associative rings. In a general perspective, strongly minimal structures, in particular the \` \` pathologies'' singled out by Hrushovski among them and contradicting Zilber's Conjecture, are still very far from being completely understood. Within modules, things are more relaxed; for instance, it is easily seen that a module \(M\) is strongly minimal if and only if every proper pp-definable subgroup is finite, in particular strong minimality just coincides with minimality in this setting. Anyhow, many interesting problems arise when classifying strongly minimal modules. As said, the paper provides a wide panorama of this matter, collecting several previous contributions together, mostly of the author, but also of others. Firstly, commutative rings are considered; here, the description of strongly minimal modules over commutative Prüfer rings is illustrated. Divisible modules over integral domains are also treated in this commutative setting; indeed, strong minimality implies a weak form of divisibility, requiring that, for any \(r\) in the underlying ring \(R\), either \(Mr = 0\) or \(Mr=M\) (and so ensuring that \(M\) is divisible over the domain \(R/\text{Ann}_R (M)\)). Then, arbitrary rings are examined, and the main results about strongly minimal modules over rings with infinite center, or over right distributive rings, or over right Ore domains are recalled.
scientific article; zbMATH DE number 1964712