Classifying E-algebras over Dedekind domains

Göbel, Rüdiger; Goldsmith, Brendan

doi:10.1016/j.jalgebra.2006.01.036

Cited by 7 publications

(3 citation statements)

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“…It follows from Theorem 7.3 that any E-ring is "the image" of the ring Z for some idempotent functor L. A torsion E-ring is a finite direct product of the rings Z p k for distinct p. The cardinalities of torsion-free E-rings are not bounded [23]; therefore, the cardinality |LZ| is not bounded. The structure of mixed E-rings can be quite difficult [27].…”

Section: Idempotent Functors and Localizations In The Category Of Abementioning

confidence: 99%

Idempotent functors and localizations in categories of modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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Section: Idempotent Functors and Localizations In The Category Of Abementioning

confidence: 99%

Idempotent functors and localizations in categories of modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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“…The class of E-rings attracted works concerning its existence and properties (see [3,5,6,9,10,15,16,17,22,24,25]). Several of these results are discussed in the monograph [20].…”

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confidence: 99%

Skeletons, bodies and generalized $E(R)$-algebras

Göbel¹,

Herden²,

Shelah³

2009

J. Eur. Math. Soc.

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In this paper we want to solve a fifty year old problem on R-algebras over cotorsionfree commutative rings R with 1. For simplicity (but only for the abstract) we will assume that R is any countable principal ideal domain, but not a field. For example R can be the ring Z or the polynomial ring Q [x]. An R-algebra A is called a generalized E(R)-algebra if its algebra End R A of R-module endomorphisms of the underlying R-module R A is isomorphic to A (as an R-algebra). Properties, including the existence of such algebras are derived in various papers ([5, 6, 9, 10, 20, 22, 24, 25]). The study was stimulated by Fuchs [13], and specially by Schultz [26]. But due to [26] the investigation concentrated on ordinary commutative E(R)-algebras. A substantial part of problem 45 (p. 232) in the monograph [13] (repeated in later publications, e.g. [27]), which will be answered positively for all rings R above in this paper, remained open:Can we find non-commutative generalized E(R)-algebras?In Theorem 1.5 we will show that for all countable, principal ideal domains R which are not fields and for any infinite cardinal κ there is a non-commutative R-algebra A of cardinality |A| = κ ℵ 0 with End R A ∼ = A, so A is a non-commutative generalized E(R)-algebra, and-not too surprisingly-there is a proper class of examples.The new strategy should be interesting and useful for other problems as well: We will first translate the heart of the algebraic question on the existence of certain monoids via model theory into geometric structures leading to a special class of (decorated) trees and solve this problem introducing products of trees etc. This can be compared with the well-known, but different process of translating group problems to small cancelations in groups via the van Kampen lemma. By small cancelation of trees we are able to find a suitable monoid and thus a non-commutative algebra A with an important non-canonical embedding A → End R A, our * -scalar multiplication. In a second part of this paper we must enlarge A to get rid of all undesired endomorphisms. This can be done more easily (thus first) with the help of additional set theory (Jensen's diamond predictions), which will support the reader to understand more quickly the last steps of the proof. In a final chapter we will also give an argument (for removing unwanted endomorphisms) which is based on R. Göbel:Mathematics Subject Classification (2000): Primary 20K20, 20K30; Secondary 16S60, 16W20 GbHSh:943 in Shelah's list of publications 846 Rüdiger Göbel et al. ordinary set theory of ZFC only (using our favored Black Box predictions; see [20]). Thus we get the result as stated above. Furthermore, this last chapter includes a construction for rigid systems of non-commutative generalized E(R)-algebras. IntroductionAs indicated in the abstract we want to construct certain non-commutative R-algebras over commutative rings R with 1. Recall that an R-algebra A is a generalized E(R)algebra if End R A ∼ = A as R-algebras, where End R M denotes the algebra of R-module endomorphi...

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“…E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decade, see [8,10,13,16,21,24,25,27,32,31]. Despite some efforts ([25, 10]) it remained an open question whether proper generalized E(R)-algebras exist.…”

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Generalized E-algebras via λ-calculus I

Göbel

Shelah

2006

Fund. Math.

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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 p n R = 0). As explained in the introduction we assume that either |R| < 2 ℵ 0 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.

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Classifying E-algebras over Dedekind domains

Abstract: An R-algebra A is said to be a generalized E-algebra if A is isomorphic to the algebra End R (A). Generalized E-algebras have been extensively investigated. In this work they are classified 'modulo cotorsion-free modules' when the underlying ring R is a Dedekind domain.

Cited by 7 publications

References 14 publications

Idempotent functors and localizations in categories of modules and Abelian groups

Idempotent functors and localizations in categories of modules and Abelian groups

Skeletons, bodies and generalized $E(R)$-algebras

Generalized E-algebras via λ-calculus I

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