Möbius transformations have been studied over the field of complex numbers. In this paper, we investigate Möbius 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 2 ( 2 R), SL 2 ( 2 R), GL 2 (D), and SL 2 (D).
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