The groups of two by two matrices in double and dual numbers, and associated Möbius transformations

Abstract: 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 $\GL{\fpower{\BR}},$ $\SL{\fpower{\BR}},$ $\GL{\BD},$ and $\SL{\BD}.$