We present a new approach for computing normal forms in the 
quotient algebra $\Ac$ of the polynomial ring $R$ by an ideal $I$. It is 
based on a criterion, which gives a necessary and sufficient condition for a 
projection onto a set of polynomials, to be a normal form modulo the ideal
$I$. 
This criterion, which does not require any monomial 
ordering, generalizes the Buchberger criterion of S-polynomials. It 
allows us to consider  more general situations than with Gr\"obner basis 
and more geometric deformations than ``initial ideal'' deformation. 
This leads to a new algorithm for constructing the multiplicative structure 
of a zero-dimensional algebra, which can apply for approximate coefficients.
It extends naturally to Laurent  polynomials and also leads to substantial
improvement in resultant based methods for solving polynomial systems, as it
will be illustrated for Macaulay matrices.