On Finite Noncommutative Gröbner 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 Gröbner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map γ from the noncommutative polynomial ring k⟨X1, …, Xn⟩ to the commutative one k[x1, …, xn] and proved that any ideal [Formula: see text] of k⟨X1, …, Xn⟩, written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn], amits a finite Gröbner basis with respect to a special monomial ordering on k⟨X1, …, Xn⟩. In this work, we approach the opposite problem. We prove that under some conditions, any ideal [Formula: see text] of k⟨X1, …, Xn⟩ admitting a finite Gröbner basis can be written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn].