LIFTINGS OF THEELEMENTARY GROUP OVER ASSOCIATIVE RINGS

Abstract

Let R be anassociative ring with 1, and let I be a nilpotent two-sidedideal of R. Assume further that there exists z \in Z(R) suchthat z, z^2-1 \in R^*. Let m \in ℕ with m \geq3. In this paper we describe all liftings of the elementary group{\tf="times-b"E}_m(R/I\,) to the general linear group{\tf="times-b"GL}_m(R), i.e. all splittings of the natural projection{\tf="times-b"E}_m(R) + {\tf="times-b"M}_m(I\,) \rightarrow{\tf="times-b"E}_m(R/I\,).