Co-generators for algebras over fields and commutative applications

Abstract

Let R be an algebra over a field k. We say that an ideal I of R is co-generated by a set F of k-linear functionals R → k if I is the largest ideal of R that is contained in the kernels of all the functionals in F;. We will state sufficient conditions for an ideal of a free k-algebra to be co-generated by finitely many functionals. We then get as a corollary that every ideal of the polynomial algebra over a field in finitely many variables is finitely co-generated. This is a known result, but the way we construct the co-generators in the general case, leads to that we get a tight bound on how many co-generators are needed for such ideals.