Multicomplex Ideals, Modules and Hilbert Spaces

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 ${\mathrm{\Lambda }}_n$, i.e. a composition of the $n$ multicomplex conjugates ${\mathrm{\Lambda }}_n:={†}_1\cdots {†}_n$, 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.