Generalized E-algebras via λ-calculus I

Abstract

An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra End R A of the R-module RA, taking any a ∈ A to the right multiplication a r ∈ End R A 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 [4]. 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 [3, 5, 7, 8, 10, 13, 14, 15, 18, 19]. Despite some efforts ([14, 5]) it remained an open question whether proper generalized E(R)-algebras exist. These are R-algebras A isomorphic to End R A but not under the above canonical isomorphism, so not E(R)-algebras. This question was raised about 30 years ago (for R - ℤ) by Schultz [21] (see also Vinsonhaler [24]). It originates from Problem 45 in Fuchs [9], that asks for a characterization of the rings A for which A ≅ End 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 ∈ R such that the additive group R + is p-torsion-free and p-reduced (equivalently p is not a zero-divisor and ∩ n∈ω P n R = 0). As explained in the introduction we assume that either |R| < 2 N0 or R + is free (see Definition 1.1). The main tool is an interesting connection between λ-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 [12]. However the general strategy will be the same, but the combinatorial arguments will utilize a prediction principle that holds under ZFC.