I have a ring (field) R, a polynomial ring $R[x_1,x_2,...,x_n]$ and a quotient module (vector space) $R[x_1,x_2,...,x_n]/I$ where $I$ is an ideal of $R[x_1,x_2,...,x_n]$ . For the case, when R is a field, the basis of the quotient vector space can be found to consist of the cosets of monomials which are not divisible by leading monomials of grobner basis of I. A similar theorem exists for the case when $R$ is a ring with effective coset representatives. For example, the ring of integers. Can I find these coset representatives that form the basis for the quotient module/vector space in sage?
I have a ring (field) R, $R$, a polynomial ring $R[x_1,x_2,...,x_n]$ and a quotient module (vector space) $R[x_1,x_2,...,x_n]/I$ where $I$ is an ideal of $R[x_1,x_2,...,x_n]$ . For the case, when R $R$ is a field, the basis of the quotient vector space can be found to consist of the cosets of monomials which are not divisible by leading monomials of grobner basis of I. $I$. A similar theorem exists for the case when $R$ is a ring with effective coset representatives. For example, the ring of integers. Can I find these coset representatives that form the basis for the quotient module/vector space in sage? sage?