  We introduce and study model-theoretic connected components of rings as an
analogue of model-theoretic connected components of definable groups. We
develop their basic theory and use them to describe both the definable and
classical Bohr compactifications of rings. We then use model-theoretic
connected components to explicitly calculate Bohr compactifications of some
classical matrix groups, such as the discrete Heisenberg group $UT_3(Z)$, the
continuous Heisenberg group $UT_3(R)$, and, more generally, groups of upper
unitriangular and invertible upper triangular matrices over unital rings.
