On Linear Functionals and Hahn-Banach Theorems for Hyperbolic and Bicomplex Modules

We consider modules over the commutative rings of hyperbolic and bicomplex numbers. In both cases they are endowed with norms which take values in non–negative hyperbolic numbers. The exact analogues of the classical versions of the Hahn–Banach theorem are proved together with some of their consequences. Linear functionals on these modules are studied and their relations with the corresponding hyperplanes are established. Finally, we introduce the notion of hyperbolic convexity for hyperbolic modules (in analogy with real, not complex, convexity) and establish its relation with hyperplanes.