Generalized E-algebras via λ-calculus I

Rüdiger Göbel, and Saharon Shelah. "Generalized E-algebras via λ-calculus I." Fundamenta Mathematicae 192.2 (2006): 155-181. <http://eudml.org/doc/282756>.

@article{RüdigerGöbel2006,
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 $_\{R\}A$, taking any a ∈ A to the right multiplication $a_\{r\} ∈ End_\{R\}A$ by a, is an isomorphism of algebras. In this case $_\{R\}A$ 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 $⋂_\{n∈ω\} pⁿR = 0$). As explained in the introduction we assume that either $|R| < 2^\{ℵ₀\}$ 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.},
author = {Rüdiger Göbel, Saharon Shelah},
journal = {Fundamenta Mathematicae},
keywords = {-rings; -algebras; endomorphism rings},
language = {eng},
number = {2},
pages = {155-181},
title = {Generalized E-algebras via λ-calculus I},
url = {http://eudml.org/doc/282756},
volume = {192},
year = {2006},
}

TY - JOUR
AU - Rüdiger Göbel
AU - Saharon Shelah
TI - Generalized E-algebras via λ-calculus I
JO - Fundamenta Mathematicae
PY - 2006
VL - 192
IS - 2
SP - 155
EP - 181
AB - 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 $_{R}A$, taking any a ∈ A to the right multiplication $a_{r} ∈ End_{R}A$ by a, is an isomorphism of algebras. In this case $_{R}A$ 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 $⋂_{n∈ω} pⁿR = 0$). As explained in the introduction we assume that either $|R| < 2^{ℵ₀}$ 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.
LA - eng
KW - -rings; -algebras; endomorphism rings
UR - http://eudml.org/doc/282756
ER -