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.

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.

In this paper we show that a ring derivation on a function algebra is trivial provided that the Choquet boundary of the algebra contains a dense sequentially non-isolated set.

Let Zm denote the ring of integers modulo an integer m > 1 and let G(Zm) be the group under composition of all Zm-automorphisms of Zm[X]. In this paper we determine Zm[X]G(zm) the subring of Zm[X] left fixed by elements of G(Zm).

If an automorphism a on a ring R (with 1) leaves no non-trivial proper ideals of R invariant then we say that R is σ-simple. We construct examples of σ-simple rings and prove that finitely generated σ-simple algebras over fields are regular. A geometric interpretation of these concepts is also discussed.