Superdecomposable pure injective modules over commutative Noetherian rings

We investigate width and Krull-Gabriel dimension over commutative Noetherian rings which are "tame" according to the Klingler-Levy approach, in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases

Strongly minimal modules over group rings

We consider modules over a group ring RG where R is a countable Dedekind domain and G is a finite group. We describe the internal structure of those RG-modules which are strongly minimal or satisfy other related model theoretic and algebraic minimality conditions

Decidability of the theory of modules over discrete valuation domains

We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all V-modules is decidable

Krull-Gabriel dimension and the model-theoretic complexity of the category of modules over group rings of finite groups

We classify group rings of finite groups over a field F according to the model-theoretic complexity of the category of their modules. For instance, we prove that, if F contains a primitive cubic root of 1, then the Krull-Gabriel dimension of such rings is 0, 2 or undefined

Superdecomposable pure-injective modules and integral group rings

We prove that, if G is a non-trivial finite group, then the integral group ring ZG possesses a superdecomposable pure-injective module

Minimalities and modules over Dedekind-like rings

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings

Minimal modules over serial rings

We investigate several model theoretic minimalities in the framework of modules over a given serial ring R