Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

An $R$-algebra $A$ is called an $E(R)$-algebra if the canonical homomorphism from $A$ to the endomorphism algebra $\mathop{\rm End}_RA$ of the $R$-module ${}_RA$, taking any $a \in A$ to the right multiplication $a_r\in \mathop{\rm End}_RA$ by $a$, is an isomorphism of algebras. In this case ${}_RA$ is called an $E(R)$-module. There is a proper class of examples constructed in \cite{DMV}. $E(R)$-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see \cite{DG2,DV,Fa,FHR,GG, GSS, GSS1,GSt,Pi,PV}. Despite some efforts (\cite{GSS1, DV}) it remained an open question whether proper generalized $E(R)$-algebras exist. These are $R$-algebras $A$ isomorphic to $\mathop{\rm End}_RA$ but not under the above canonical isomorphism, so not $E(R)$-algebras. This question was raised about 30 years ago (for $R=\mathbb Z$) by Schultz \cite{Sch} (see also Vinsonhaler \cite{Vi}). It originates from Problem 45 in Fuchs \cite{Fu0}, that asks for a characterization of the rings $A$ for which $A\cong \mathop{\rm End}_\mathbb Z A$ (as rings). We answer Schultz's question, thus contributing a large class of rings for Fuchs' Problem 45 which are not $E$-rings. Let $R$ be a commutative ring with an element $p\in R$ such that the additive group $R^+$ is $p$-torsion-free and $p$-reduced (equivalently $p$ is not a zero-divisor and $\bigcap_{n\in\omega} p^nR=0$). As explained in the introduction we assume that either $|R| <2^{\aleph_0}$ or $R^+$ is free (see Definition 1.1).

The main tool is an interesting connection between $\lambda$-calculus (used in theoretical computer science) and algebra. It seems reasonable to divide the work into two parts; in this paper we work in $ V= L$ (Gödel's universe) where stronger combinatorial methods make the final arguments more transparent. The proof based entirely on ordinary set theory (the axioms of ZFC) will appear in a subsequent paper \cite{GS}. However the general strategy will be the same, but the combinatorial arguments will utilize a prediction principle that holds under ZFC.