The groups of two by two matrices in double and dual numbers, and associated M\"obius transformations

M"obius transformations have been studied over the field of complex numbers. In this paper, we investigate M"obius transformations over two rings which are not fields: the ring of double numbers and the ring of dual numbers. We give types of continuous one-parameter subgroups of $\text{GL}\text{fpower}\text{BR},$ $\text{SL}\text{fpower}\text{BR},$ $\text{GL}\text{BD},$ and $\text{SL}\text{BD}\mathrm{.}$