A note on the π-complemented algebras

Juan Carlos Cabello

Juan Carlos Cabello Departamento de Analisis Matematico. Facultad de Ciencias, University of Granada, Spain

Keywords: semiprime algebra, extended centroid, central closure, complemented algebra

Abstract

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. Let B_A denote the set of all idempotents of the extended centroid of a semiprime algebra A. We prove that if there is a maximal ideal P of B_A such that PA ⊆ A then A and PA are two π-complemented algebras. As a consequence, we give a characterization of the π-complementation of the unitisation, and the multiplication ideal, of a semiprime algebra.

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