For a finite abelian group A, the Reidemeister number of an endomorphism φ is the same as the number of fixed points of φ, and the Reidemeister spectrum of A is completely determined by the Reidemeister spectra of its Sylow p-subgroups. To compute the Reidemeister spectrum of a finite abelian p-group P, we introduce a new number associated to an automorphism ψ of P that captures the number of fixed points of ψ and its (additive) multiples, we provide upper and lower bounds for that number, and we prove that every power of p between those bounds occurs as such a number.

A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

Let $M$ be a module over the ring $R$. Extensive use is made of Krull codimension to study further the Artinian-finitary automorphism group $\[ F_1\hbox{\rm Aut}_RM \,{=}\, \{g\,{\in}\hbox{ Aut}_RM \,{:}\, M(g - 1) \hbox{ is } R\hbox{-Artinian}\} \]$ of $M$ over $R$. Substantial progress is made where either $M$ is residually Noetherian or $R$ is commutative. There are some group-theoretic consequences of the two main structure theorems.

An $E$-ring is a unital ring $R$ such that every endomorphism of the underlying abelian group ${{R}^{+}}$ is multiplication by some ring element. The existence of almost-free $E$-rings of cardinality greater than ${{2}^{{{\aleph }_{0}}}}$ is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ${{\aleph }_{1}}\le \text{ }\!\!\lambda\!\!\text{ }\le {{2}^{{{\aleph }_{0}}}}$ we construct $E$-rings of cardinality $\lambda $ in ZFC which have ${{\aleph }_{1}}$-free additive structure. For $\text{ }\!\!\lambda\!\!\text{ }={{\aleph }_{1}}$ we therefore obtain the existence of almost-free $E$-rings of cardinality ${{\aleph }_{1}}$ in ZFC.

Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an ${{\aleph }_{1}}$-free $R$-module $G$ of rank ${{\aleph }_{1}}$ with endomorphism algebra $\text{En}{{\text{d}}_{R}}\,G=A$. Clearly the result does not hold for fields. Recall that an $R$-module is ${{\aleph }_{1}}$-free if all its countable submodules are free, a condition closely related to Pontryagin’s theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A\,=\,R$, then clearly $G$ is an indecomposable ‘almost free’module. The existence of such modules was unknown for rings with only finitelymany primes like $R={{\mathbb{Z}}_{\left( p \right)}}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner’s showing that any such algebra is an endomorphism algebra of some torsionfree, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorialalgebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.