One-parameter Groups of Möbius Maps in Two-Dimensional Real Commutative Algebra

Möbius transformations have been thoughtfully studied over the field of complex numbers. In this thesis, we investigate Möbius transformations over two rings which are not fields: the ring of double numbers $\mathbb O$ and the ring of dual numbers $\mathbb D$. We will see certain similarity between the cases of fields and rings along with some significant distinctions. After the introduction and necessary background material, given in the first two chapters, I introduce general linear groups, projective lines and Möbius transformations over several rings such us the ring of integer numbers, the Cartesian product ring and the two rings $\mathbb O$ and $\mathbb D.$ In the following chapters, we consider in details metrics, classification of Möbius maps based on the number of fixed points, connected continuous one-parameter subgroups and an application of Möbius maps.