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 $$\Lambda _n$$ , 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.