  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 $\SL{\BD}.$
