Multicomplex Ideals, Modules and Hilbert Spaces

In this article we study some algebraic aspects of multicomplex numbers Mₙ. A canonical idempotent representation defined in terms of n multicomplex conjugates is introduced. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy, i.e. a composition of the n multicomplex conjugates, as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied. Multicomplex free modules and their linear operators are introduced. Finally, we develop multicomplex Hilbert spaces.