Multicomplex Ideals, Modules and Hilbert Spaces

Abstract

In this article we study some algebraic aspects of multicomplex numbers \({\mathbb {M}}_n\). For \(n\ge 2\) a canonical representation is defined in terms of the multiplication of \(n-1\) idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy \(\Lambda _n\), i.e. a composition of the n multicomplex conjugates \(\Lambda _n:=\dagger _1\cdots \dagger _n\), as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied in details, free \({\mathbb {M}}_n\)-modules and their linear operators are considered and, finally, we develop Hilbert spaces on the multicomplex algebra.