Nil subrings of endomorphism rings of finitely generated modules over affine PI-rings.
type: Item
instance of: scholarly article
title: Nil subrings of endomorphism rings of finitely generated modules over affine PI-rings.
zbMATH Open document ID: 1218.16014
author: Robert Guralnick
author: Lance W. Small
author: E. Zelmanov
published in: Journal of Algebra
publication date: 2011-02-18T00:00:00Z
review text: The authors prove the following result. Let \(R\) be an affine PI-algebra over a commutative Noetherian ring \(C\). Let \(M_R\) be a finitely generated \(R\)-module. Then any nil subring of \(\text{End}_R(M)\) is nilpotent. The authors point out the following two corollaries of this result. Let \(R\) be an affine PI-algebra over a commutative Noetherian ring \(C\), and let \(M_R\) be a finitely generated \(R\)-module. If \(V\) is a nil subset of \(\text{End}_R(M)\) closed under the Lie bracket, then \(V\) generates a nilpotent ring. To formulate the second corollary we recall that a Lie ring \(L\) is said to be weakly Engel if given \(x\) from \(L\) there exists a positive integer \(n(x)\) such that \(\text{ad}(x)^{n(x)}(y)=0\) for any \(y\in L\). The second corollary asserts that for any affine PI-algebra \(A\) over a commutative Noetherian ring \(C\) if \(L\) is a weakly Engel Lie subring of \(A\), then \(L\) is a nilpotent Lie ring. Other corollaries are pointed out.
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 5853360
zbMATH Keywords: nil rings
zbMATH Keywords: affine PI-rings
zbMATH Keywords: endomorphism rings
zbMATH Keywords: nilpotency
zbMATH Keywords: nilpotent rings
zbMATH Keywords: Lie subrings
zbMATH Keywords: weakly Engel Lie rings
zbMATH Keywords: nilpotent ideals
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.06.005
OpenAlex ID: W1994705429
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cites work: Q5602178
cites work: Associative rings satisfying the Engel condition
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cites work: On associative algebras satisfying the Engel condition
DOI: 10.1016/J.JALGEBRA.2010.06.005
description: scientific article; zbMATH DE number 5853360