Algebraic Structure and Classification of Idempotents in Multicomplex Spaces C_n

In this paper, we introduce the multicomplex space C_n generated by n commuting imaginary units and develop its fundamental algebraic framework. We define the algebraic operations on C_n and establish that it possesses a natural structure of a modified Banach algebra. A systematic study of idempotent elements in C_n is carried out, including their explicit construction, classification into trivial, primitive, and compound types, and their organization according to levels. Furthermore, we investigate the norm structure of idempotent elements and determine all possible norm values, providing explicit formulas and low-dimensional illustrations. The results unify the algebraic and metric aspects of multicomplex algebras and offer a clear structural understanding of idempotents in C_n, which may be useful for further developments in multicomplex analysis and functional algebra.