  We study Dirac operators acting on sections of a Clifford module ${\cal E}$\
over a Riemannian manifold $M$. We prove the intrinsic decomposition formula
for their square, which is the generalisation of the well-known formula due to
Lichnerowicz [L]. This formula enables us to distinguish Dirac operators of
simple type. For each Dirac operator of this natural class the local
Atiyah-Singer index theorem holds. Furthermore, if $M$\ is compact and
${{\petit \rm dim}\;M=2n\ge 4}$, we derive an expression for the Wodzicki
function $W_{\cal E}$, which is defined via the non-commutative residue on the
space of all Dirac operators ${\cal D}({\cal E})$. We calculate this function
for certain Dirac operators explicitly. From a physical point of view this
provides a method to derive gravity, resp. combined gravity/Yang-Mills actions
from the Dirac operators in question.
