We present a proof of embedded desingularization for closed
subschemes which does not make use of Hilbert-Samuel function and
avoids Hironaka's notion of normal flatness (see also
\cite{EncinasVillamayor2000} page 224). Given a subscheme defined
by equations, we prove that embedded desingularization can be
achieved by a sequence of monoidal transformations; where the law
of transformation on the equations defining the subscheme is
simpler then that used in Hironaka's procedure.
This is done by showing that desingularization of a closed
subscheme $X$, in a smooth sheme $W$, is achieved by taking an
algorithmic principalization for the ideal $I(X)$, associated
to the embedded scheme $X$. This provides a conceptual
simplification of the original proof of Hironaka. This algorithm
of principalization (of Log-resolution of ideals), and this new
procedure of embedded desingularization discussed here, have been
implemented in MAPLE.