Left rigid rings.
type: Item
instance of: scholarly article
title: Left rigid rings.
zbMATH Open document ID: 1122.16023
author: Joshua Buckner
author: Manfred Dugas
published in: Journal of Algebra
publication date: 2007-04-11T00:00:00Z
review text: Let \(R\) be a ring with \(1\) and \(\mathcal F\) a family of left ideals of \(R\). Then, by definition, \(\text{End}(R^+,\mathcal F)=\{\varphi\in\text{End}(R^+)\mid\forall\,X\in\mathcal F:\varphi(X)\subseteq X\}\). The authors ask when \(\text{End}(R^+,\mathcal F)=R\) and they call \(R\) \(\mathcal F\)-rigid if this is the case. Of particular interest is the case when \(\mathcal F=\mathcal F_\ell\), the family of all left ideals of \(R\). There is no complete solution to this question but there are numerous examples and special cases. Sample results are as follows. Theorem~1. Let \(S\) be a ring with identity, \(\kappa\) a cardinal \(\geq 2\), \(F\) a free \(S\)-module of rank \(\kappa\). Let \(R=\text{End}_S(F)\). Then there exist families \(\mathcal F\), \(\mathcal F_{fin}\) and \(\mathcal F_S\) of left ideals of \(R\) such that (1) \(\text{End}(R^+,\mathcal F)=R\), and if \(\kappa\) is finite, (2) \(\text{End}(R^+,\mathcal F_{fin})=R\), and (3) \(\text{End}_S(R,\mathcal F_S)=R\). Corollary~1. If \(F\) is a field and \(R=\text{End}(F^+)\), then \(\text{End}(R^+,\mathcal F_\ell)=R\). Theorem~4. Let \(R\) be a PID with only finitely many maximal ideals and having a non-zero derivation. Then there is no family \(\mathcal F\) such that \(R=\text{End}(R^+,\mathcal F)\). Corollary~3. Let \(K\) be a field and \(P_1,\dots,P_n\) finitely many maximal ideals of the polynomial ring \(K[x]\). Let \(R\) be the localization of \(K[x]\) at the ideals \(P_i\). Then there is no family \(\mathcal F\) such that \(\text{End}(R^+,\mathcal F)=R\). The authors also deal with polynomial rings in an arbitrary number of indeterminates and with subrings of algebraic number fields. A sample result of the latter is Corollary~5. Let \(S\) be an integrally closed full subring of the finite Galois extension \(F\) of \(\mathbb{Q}\) and \(\mathcal F\) a family of left ideals of \(S\). Then \(E_{\mathcal F}=\text{End}(S,\mathcal F)=S\) or else \(\text{End}(E_{\mathcal F},\widetilde F)=E_{\mathcal …
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 5141714
zbMATH Keywords: additive groups of rings
zbMATH Keywords: endomorphisms of additive groups
zbMATH Keywords: families of left ideals
zbMATH Keywords: maximal ideals
zbMATH Keywords: polynomial rings
zbMATH Keywords: localizations
zbMATH Keywords: Abelian groups
zbMATH Keywords: endomorphism rings
reviewed by: Adolf Mader
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.10.017
OpenAlex ID: W2029420279
cites work: Differentially trivial and rigid rings of finite rank
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DOI: 10.1016/J.JALGEBRA.2006.10.017
description: scientific article; zbMATH DE number 5141714