  We introduce a class of rings, namely the class of left or right $p$-nil
rings, for which the adjoint groups behave regularly. Every $p$-ring is close
to being left or right $p$-nil in the sense that it contains a large ideal
belonging to this class. Also their adjoint groups occur naturally as groups of
automorphisms of $p$-groups. These facts and some of their applications are
investigated in this paper.
