Postmodern Algebra

Suppose \(R=R_0[f_1,\dots,f_n]\text{.}\) Any element \(r\in R_+\) can be written as a polynomial expression \(r=P(f_1,\dots,f_n)\) for some \(P\in R_0[x_1,\dots,x_n]\) with no constant term. Each monomial of \(P\) is a multiple of some \(x_i\text{,}\) and thus each term in \(r=P(f_1,\dots,f_n)\) is a multiple of \(f_i\text{.}\) Thus \(r \in R f_1 + \cdots + R f_n = (f_1,\dots,f_n)\text{.}\)

To show that \(R_+= (f_1,\dots,f_n)\) implies \(R=R_0[f_1,\dots,f_n]\text{,}\) it suffices to show that any homogeneous element \(r\in R\) can be written as a polynomial expression in \(f_1, \ldots, f_n\) with coefficients in \(R_0\text{.}\) We will use induction on the degree of \(r\text{,}\) with degree \(0\) as a trivial base case. For \(r\) homogeneous of positive degree, we must have \(r\in R_+\text{,}\) so by assumption we can write \(r= a_1 f_1 + \dots + a_n f_n\text{.}\) Moreover, since \(r\) and \(f_1, \ldots, f_n\) are all homogeneous, we can choose each coefficient \(a_i\) to be homogeneous of degree \(|r|-|f_i|\text{.}\) By the induction hypothesis, each \(a_i\) is a polynomial expression in \(f_1, \ldots, f_n\text{,}\) so we are done.

If \(R_0\) is noetherian and \(R\) is algebra-finite over \(R_0\text{,}\) then \(R\) is noetherian by the Hilbert’s Basis Theorem. On the other hand, if \(R\) is noetherian then any quotient of \(R\) is also noetherian, and in particular \(R_0 \cong R/R_+\) is noetherian. Moreover, \(R_+\) is generated as an ideal by finitely many homogeneous elements by noetherianity; by [provisional cross-reference: cite] [[Mathematics/Commutative Algebra/Results/Proposition - Graded F.G. Algebra|Proposition]], we get a finite algebra generating set for \(R\) over \(R_0\text{.}\)