Classification theory for abelian groups with an endomorphism
The paper considers classifiability for Abelian groups with an endomorphism x, hence for Z[x]-modules. In stability theory, classifying the models of some complete theory means to determine their isomorphism types by some kind of invariant (like a ``short'' sequence of cardinals). When the models are modules over some fixed countable ring R, this assignment of invariants can be done if and only if the theory is superstable. Moreover there are good reasons to believe that, if the ring R is of wild representation type, then this classification is strongly complicated, even if not compromised. The authors first provide a full characterization of superstable K[x]- modules where K is any countable field (notice that this includes the case \(K={\mathbb{Q}}\), hence Z[x]-modules which are divisible and torsionfree over Z). Secondly they consider Z[x]-modules whose underlying group has a finite exponent m, hence Z/m[x]-modules. Here it is shown that in most cases Z/m[x] and its quotient rings are of wild representation type, and the corresponding theories of modules are undecidable. The same happens for Z[x]-modules which are divisible (or torsionfree) as Z-modules. A parallel analysis is given when x is an automorphism, so for \(Z[x,x^{- 1}]\)-modules.
scientific article; zbMATH DE number 4191608