Let R be a unital commutative ring and A a unital R-algebra. We introduce the category of E(A,R)-modules which is a natural extension of the category of E-modules. The properties of E(A,R)-modules are studied; in particular we consider the subclass of E(R)-algebras. This subclass is of special interest since it coincides with the class of E-rings in the case R = ℤ. Assuming diamond ⋄, almost-free E(R)-algebras of cardinality κ are constructed for any regular non-weakly compact cardinal κ > ℵ...

Let R be a subring of the rational numbers ℚ. We recall from [3] that an R-module G is a splitter if $Ext{¹}_R(G,G)=0$. In this note we correct the statement of Main Theorem 1.5 in [3] and discuss the existence of non-free splitters of cardinality ℵ₁ under the negation of the special continuum hypothesis CH.

Let R be a ring with identity such that R⁺, the additive group of R, is torsion-free. If there is some R-module M such that $R\subseteq M\subseteq \mathbb{Q}R(=\mathbb{Q}{\otimes }_{\mathbb{Z}}R)$ and $En{d}_{\mathbb{Z}}\left(M\right)=R$, we call R a Zassenhaus ring. Hans Zassenhaus showed in 1967 that whenever R⁺ is free of finite rank, then R is a Zassenhaus ring. We will show that if R⁺ is free of countable rank and each element of R is algebraic over ℚ, then R is a Zassenhaus ring. We will give an example showing that this restriction on R is needed. Moreover, we will show that a ring due to A....