Combinatorial extensions of radical, nil, and locally nilpotent algebras
type: Item
instance of: scholarly article
title: Combinatorial extensions of radical, nil, and locally nilpotent algebras
zbMATH Open document ID: 0572.16003
DOI: 10.1016/0021-8693(85)90082-1
author: Hans-Christian Mez
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let K be a commutative field. The author shows that every radical K- algebra (i.e. coinciding with its Jacobson radical) can be embedded in a radical K-algebra A with the property that A contains no nonzero ideal which is nil or residually nilpotent or satisfies a polynomial identity. The construction is analogous to that used by \textit{L. A. Bokut'} [Sib. Mat. Zh. 4, 500-518 (1963; Zbl 0136.022)] to prove that for any \(a\neq 0\) and b, two elements u, v can be adjoined to satisfy \(b=uav\). Secondly, define an algebra to be m-nilpotent if each subalgebra generated by m elements is nilpotent. Then the author shows that every locally nilpotent (m-nilpotent) K-algebra can be embedded in a locally nilpotent (resp. m- nilpotent) K-algebra A containing no nonzero ideal which is nil of bounded index or satisfies a polynomial identity. Using methods of Golod, he deduces that every nil K-algebra can be embedded in a nil K-algebra whose locally nilpotent ideals are totally ordered by inclusion.
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16Nxx
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16Dxx
zbMATH DE Number: 3912552
zbMATH Keywords: radical K-algebra
zbMATH Keywords: Jacobson radical
zbMATH Keywords: residually nilpotent
zbMATH Keywords: polynomial identity
zbMATH Keywords: locally nilpotent
zbMATH Keywords: m-nilpotent
zbMATH Keywords: nil K-algebra
zbMATH Keywords: locally nilpotent ideals
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90082-1
OpenAlex ID: W2033628719
cites work: ON ELEMENTARY PROPERTIES OF EXISTENTIALLY CLOSED SYSTEMS
cites work: Embeddings into simple associative algebras
cites work: Q5639812
cites work: Q5515373
cites work: Corrections to: 'Free ideal rings'
cites work: Q5513080
cites work: On nil-algebras and finitely approximable 𝑝-groups
cites work: Q4339908
cites work: Forcing, arithmetic, division rings
cites work: Q3342725
cites work: Q5731846
cites work: Q5622720
cites work: An example of a simple radical ring
cites work: Q3910643
cites work: The ideal structure of existentially closed algebras
Recommended article: On Nilpotent Extensions of Algebras
Recommended article: Q3676302
Recommended article: A CLASS OF LOCALLY NILPOTENT COMMUTATIVE ALGEBRAS
Recommended article: Combinatorial dimensions: Indecomposability on certain local finite-dimensional trivial extension algebras
Recommended article: Q4237290
Recommended article: Nilpotent elements and nil-radicals of alternative algebras
Recommended article: Q4095021
Recommended article: Radicals and nilpotent elements of free alternative algebras
Recommended article: Q5267626
Recommended article: On local-extensions of algebras
description: scientific article; zbMATH DE number 3912552