  The gravitating matter is studied within the framework of the non-commutative
geometry. The non-commutative Einstein-Hilbert action on the product of a four
dimensional manifold with a discrete space gives the models of matter fields
coupled to the standard Einstein gravity.The matter multiplet is encoded in the
Dirac operator which yields the representation of the algebra of the universal
forms. The general form of the Dirac operator depends on a choice of the
grading of the corresponding spinor bundle. A choice is given, which leads to
the nonlinear vector sigma-model coupled to the Einstein gravity.
