Let R be a field and d R[X] of degree n > 0. The unique representatives of degree less than n in R[X]/(d) form a subspace R[X]<n of the vector space R[X]. A complement is formed by (d):

Theorem

The ring R[X] has the following vector space decomposition:

R[X] = R[X]<n dR[X].

Furthermore, the map

R[X] -> R[X]<n,    f -> f mod d

is the linear projection onto R[X]<n with kernel dR[X].