The R-submodule of R[X] consisting of polynomials of degree ≤ n.

Given a polynomial, return the polynomial whose coefficients are in the ring closure of the original coefficients.

Given a polynomial p and a subring T that contains the coefficients of p, return the corresponding polynomial whose coefficients are in `T.

Given a polynomial whose coefficients are in some subring, return the corresponding polynomial whose coefificents are in the ambient ring.

Transport an ideal of R[X] to an R-submodule of R[X].

Given an ideal I of R[X], make the R-submodule of I consisting of polynomials of degree ≤ n.

Given an ideal I of R[X], make the ideal in R of leading coefficients of polynomials in I with degree ≤ n.

Given an ideal I in R[X], make the ideal in R of the leading coefficients in I.

Hilbert basis theorem.