Splitters and relative homological algebra
type: Item
instance of: scholarly article
title: Splitters and relative homological algebra
zbMATH Open document ID: 0790.18008
DOI: 10.1016/0021-8693(92)90168-L
author: Aboubakr Lbekkouri
author: Mark Kleiner
published in: Journal of Algebra
publication date: 1993-03-10T00:00:00Z
review text: Let \(K\subset\Delta\subset\Lambda\) be rings. The subring \(\Delta\) is called a splitter of \(\Lambda\) over \(K\) if the multiplication map \(m_ K: \Lambda\otimes_ K \Lambda\to \Lambda\) has a right inverse which is a homomorphism of \(\Delta\)-bimodules. \(\Delta\) is said to be non-splittable over \(K\) if \(K\) is the only splitter of \(\Lambda\) over \(K\). The authors give several characterizations of splitters -- one most commonly used being that \(\Delta\) is a splitter of \(\Lambda\) over \(K\) if and only if there exists an \(e\in \Lambda\otimes_ K \Lambda\) such that \(m_ K(e)=1\) and \(e\delta= \delta e\) for all \(\delta\in \Delta\). Among many general properties of splitters, it is proved that when \(K\) is a field and \(\Lambda\) is a \(K\)-algebra then every splitter is a finite- dimensional subalgebra of \(\Lambda\). If, in addition, \(\Lambda\) is local, it is shown that every splitter is imbedded in a division subalgebra of \(\Lambda\). When \(K\) is an arbitrary field, \(L\) an extension of \(K\), and \(F\) the algebraic closure of \(K\) in \(L\), then \(L\) is non-splittable over \(K\) if and only if \(F\) is a purely inseparable extension of \(K\). Using this it is established that a \(K\)-algebra \(\Lambda\) is algebraic non- splittable if and only if the Jacobson radical \(J(\Lambda)\) of \(\Lambda\) is nil and \(\Lambda/ J(\Lambda)\) is a purely inseparable field extension of \(K\). The question whether an algebraic non-splittable algebra remains non- splittable under an extension of the ground field is also investigated. This question is shown to be equivalent to the classical question of whether every nil algebra remains nil under extension of the ground field. A \(K\)-algebra \(\Lambda\), \(K\) a field, is said to be totally splittable if every finite-dimensional \(K\)-subalgebra of \(\Lambda\) is a splitter. It is then shown that an algebraic field extension \(L\) of a field \(K\) is separable if and only if \(L\) is a totally splittab…
Mathematics Subject Classification ID: 18G25
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16E99
Mathematics Subject Classification ID: 12F05
zbMATH DE Number: 130528
zbMATH Keywords: multiplication map
zbMATH Keywords: non-splittable
zbMATH Keywords: splitter
zbMATH Keywords: algebraic closure
zbMATH Keywords: purely inseparable field extension
zbMATH Keywords: nil algebra
reviewed by: Lekh Raj Vermani
MaRDI profile type: Publication
cites work: Q5647010
cites work: Q3903156
cites work: Foundations of relative homological algebra
cites work: Relative Hochschild cohomology, rigid algebras, and the Bockstein
cites work: On semisimple extensions and separable extensions over non commutative rings
cites work: Relative Homological Algebra
cites work: Integrations and cohomology theory
cites work: Separable ring extensions and cohomological dimension
cites work: Q5343374
cites work: Cohomology of Infinite Algebras
cites work: Purely inseparable algebras
full work available at URL: 0021-8693(92)90168-l
OpenAlex ID: W2060244608
description: scientific article; zbMATH DE number 130528