We will introduce very briefly the homogenization process in the commutative case, just to explain the main motivation of our work. In the commutative context, the Buchberger Algorithm give us a very direct strategy for computing Grobner Basis for a given ideal I ∈ k[x1, x2, . . . , xn]: we consider a finite set {f1, f2, ...fk} of generators of I, compute the S polynomials, for any pair i,j, reduce them, and if the remainder is non zero, add this remainder to the list of the given polynomials, to make all the S polynomials reduce to zero. Although this process always finish, in the commutative case, it can be very inefficient and time consuming, by instance getting S polynomials of much higher degree that the ones we begin with. It is easy to see ( see [1]) that if we begin with a set of homogeneos polynomials this problem does not occur and the S polynomials we obtain are again homogeneous. So, lets define this process for Λ = k[x1, x2, . . . , xn]: Let f ∈ Λ and w a new variable. If f has total degree d then the polynomial given by f = wf(x1/w, x2/w, . . . , xn/w) ∈ k[x1, x2, . . . , xn, w] is a homogeneous polynomial in the extended polynomial algebra, called the homogenization of f . For an ideal I ∈ k[x1, x2, . . . , xn] define I to be the ideal of k[x1, x2, . . . , xn, w] given by I ∗ = . For any h ∈ k[x1, x2, . . . , xn, w], define h∗ = h(x1, x2, . . . , xn, 1) ∈ k[x1, x2, . . . , xn]. As we will prove, in the last section, if G is a Grobner basis for I with respect to a certain order, then the set G = {g|g ∈ G} is a Grobner Basis for the ideal I with respect to the extended order.