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π-Complemented Algebras Through Pseudocomplemented Lattices

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Abstract

For an ideal I of a nonassociative algebra A, the π-closure of I is defined by \(\overline{I} = {\rm Ann}({\rm Ann} (I))\), where Ann(I) denotes the annihilator of I, i.e., the largest ideal J of A such that IJ = JI = 0. An algebra A is said to be π-complemented if for every π-closed ideal U of A there exists a π-closed ideal V of A such that A = U ⊕ V. For instance, the centrally closed semiprime ring, and the AW ∗-algebras (or more generally, boundedly centrally closed C ∗-algebras) are π-complemented algebras. In this paper we develop a structure theory for π-complemented algebras by using and revisiting some results of the structure theory for pseudocomplemented lattices.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Juan Carlos Cabello & Miguel Cabrera

  2. Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, 29071, Málaga, Spain

    Antonio Fernández López

Authors
  1. Juan Carlos Cabello
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  2. Miguel Cabrera
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  3. Antonio Fernández López
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Corresponding author

Correspondence to Antonio Fernández López.

Additional information

J.C. Cabello and M. Cabrera were supported by the MICINN and Fondos FEDER, MTM2009-12067, and, in addition, by the Junta de Andalucía Grant FQM290. A. Fernández López was supported by the MEC and Fondos FEDER, MTM2007-61978 and MTM2010-19482.

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Cabello, J.C., Cabrera, M. & Fernández López, A. π-Complemented Algebras Through Pseudocomplemented Lattices. Order 29, 463–479 (2012). https://doi.org/10.1007/s11083-011-9214-4

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