Structure theory for multiplicatively semiprime algebras
type: Item
instance of: scholarly article
title: Structure theory for multiplicatively semiprime algebras
zbMATH Open document ID: 1124.17002
published in: Journal of Algebra
publication date: 2004-11-29T00:00:00Z
review text: The authors develop a structure theory for multiplicative semiprime algebras. If \(A\) is a non-necessarily associative algebra and \(M(A)\) is the subalgebra of \(\text{End}(A)\) generated by the identity and all left and right multiplication operators, we say that \(A\) is multiplicatively semiprime (prime) if both \(A\) and \(M(A)\) are semiprime (prime) algebras. In a preliminary section the authors study the (right) quasi-multiplicaive lattice with closure map \(\varepsilon\) with \(\varepsilon\)-right continuous (\(\varepsilon\)-continuous) product, and Galois connection between complete lattices. This result is applied to the particular cases of the lattice of ideals of an algebra \(A\) and the lattice of ideals of its algebra of multiplication, denoted by \({\mathcal I}(A)\) and \({\mathcal I}(M(A))\) respectively. The authors introduce and study the Galois connection between \({\mathcal I}(A)\) and \({\mathcal I}(M(A))\) given by the pair of maps \(I\to I^{\text{ann}}\) and \({\mathcal P}\to {\mathcal P}_{\text{ann}}\) where \[ I^{\text{ann}}:=\{F\in M(A)\mid F(I)=0\}\quad\text{and}\quad {\mathcal P}_{\text{ann}}:=\{a\in A\mid {\mathcal P}(a)=0\}. \] This connection gives rise to the notions of closed ideals of \(A\) and \(M(A)\) respectively: An ideal \(I\) of \(A\) is \(\varepsilon\)-closed if \(I=(I^{\text{ann}})_{\text{ann}}\), and ideal \(\mathcal P\) of \(M(A)\) is \(\varepsilon'\)-closed if \({\mathcal P}=({\mathcal P}_{\text{ann}})^{\text{ann}}\). Moreover, by a general argument of lattice theory they prove that the set \({\mathcal L}_\varepsilon\) of all \(\varepsilon\)-closed ideals of \(A\) (respectively, the set \({\mathcal L}_{\varepsilon'}\) of all \(\varepsilon'\)-closed ideals of \(M(A)\)) is a complete lattice with the operations \[ \bigwedge I_\lambda= \bigcap I_\lambda\quad \text{and}\quad \bigvee I_\lambda=((\sum I_\lambda)^{\text{ann}})_{\text{ann}}. \] Moreover, the natural product of \({\mathcal I}(A)\) (resp. \({\mathcal I}(M(A))\)) …
Mathematics Subject Classification ID: 17A60
Mathematics Subject Classification ID: 06D15
zbMATH DE Number: 2119569
zbMATH Keywords: multiplication algebra
zbMATH Keywords: multiplicatively semiprime
zbMATH Keywords: Galois connection
zbMATH Keywords: pseudocomplemented lattices
zbMATH Keywords: normed algebras
author: J. C. Cabello
author: M. Cabrera
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.07.035
OpenAlex ID: W2065065093
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DOI: 10.1016/J.JALGEBRA.2004.07.035
description: scientific article; zbMATH DE number 2119569