We generalize derived equivalences for triangular matrix rings induced by a certain type of classical tilting module introduced by Auslander, Platzeck and Reiten to generalize reflection functors in the representation theory of quivers due to Bernstein, Gelfand and Ponomarev.

Abstract We introduce and study model-theoretic connected components of rings as an analogue definable groups. develop their basic theory use them to describe both the classical Bohr compactifications rings. then explicitly calculate some matrix groups, such discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$ , continuous {R}})$ and, more generally, groups upper unitriangular invertible...