In a first part, we introduce *normalized rewriting*, a new
rewrite relation. It generalizes former notions of rewriting modulo a
set of equations *E*, dropping some conditions on
*E*. For example, *E* can now be the theory of identity,
idempotence, the theory of Abelian groups or the theory of commutative
rings. We give a new completion algorithm for normalized rewriting. It
contains as an instance the usual AC completion algorithm, but also
the well-known Buchberger algorithm for computing Gröbner bases of
polynomial ideals.

In a second part, we investigate the particular case of completion of
ground equations. In this case we prove by a uniform method that
completion modulo *E* terminates, for some interesting theories
*E*. As a consequence, we obtain the decidability of the word
problem for some classes of equational theories, including the
AC-ground case (a result known since 1991), the ACUI-ground case (a
new result to our knowledge), and the cases of ground equations modulo
the theory of Abelian groups and commutative rings, which is already
known when the signature contains only constants, but is new
otherwise.

Finally, we give implementation results which show the efficiency of normalized completion with respect to completion modulo AC.