A Contribution to the Study of a Class of Noncommutative Ideals Admitting Finite Gröbner Bases

Considering a field \(\mathbb {K}\) of characteristic 0, the n-variate commutative polynomial ring \(\mathbb {K}[x_1,\ldots ,x_n]\) over \(\mathbb {K}\) , the n-variate noncommutative polynomial ring \(\mathbb {K}\langle X_1,\ldots ,X_n\rangle \) over \(\mathbb {K}\) , and \(\gamma :\mathbb {K}\langle X_1,\ldots ,X_n\rangle \longrightarrow \mathbb {K}[x_1,\ldots ,x_n]\) the application sending \(X_i\) to \(x_i\) , Eisenbud et al. proved that for any ideal \(\mathcal {I}\) of \(\mathbb {K}[x_1,\ldots ,x_n]\) , the ideal \(\mathcal {J}=\gamma ^{-1}(\mathcal {I})\) has a finite Gröbner basis. Y. Diop and D. Sow dealt with the opposite problem and proved that any noncommutative ideal which contains all commutators and has a finite Gröbner basis is a preimage of a commutative ideal by \(\gamma \) . In this work, we prove that this application \(\gamma \) can be replaced by any surjective homomorphism. Thus we generalize the two results previously cited.