On Finite Noncommutative Grobner Bases

It is well known that in the noncommutative polynomial ring in serveral variables Buchberger's algorithm does not always terminate. Thus, it is important to characterize noncommutative ideals that admit a finite Grobner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map gamma from the noncommutative polynomial ring k < X-1, ..., X-n > to the commutative one k[x(1), ..., x(n)] and proved that any ideal J of k < X-1, ..., X-n >, written as J = gamma(-1)(I) for some ideal I of k[x(1), ..., x(n)], amits a finite Grobner basis with respect to a special monomial ordering on k < X-1, ..., X-n >. In this work, we approach the opposite problem. We prove that under some conditions, any ideal J of k < X-1, ..., X-n > admitting a finite Grobner basis can be written as J = gamma(-1)(I) for some ideal I of k[x(1), ..., x(n)].