Proposition 1

Let $A = \bigoplus_{i \in \mathbf{Z}} A_i$ be a graded commutative ring with positive degrees, m the graded ideal $\bigoplus_{i \geq 1} A_i$ and $(x_\lambda)_{\lambda \in L}$ a family of homogeneous elements of $A$ of degrees $\geq 1$. The following conditions are equivalent:

(a) The ideal of $A$ generated by the family $(x_\lambda)$ is equal to $m$.

(b) The family $(x_\lambda)$ is a system of generators of the $A$*-algebra* $A$.

(c) For all $i \geq 0$, the $A$*-module* $A_i$ is generated by the elements of the form $\prod_\lambda x_\lambda^{n_\lambda}$ which are of degree $i$ in $A$.

Clearly conditions (b) and (c) are equivalent. If they hold, every element of m is of the form $f((x_i))$ where $f$ is a polynomial of $A_0[X_\lambda]_{\lambda \in L}$ with no constant term; then $m = \sum_{\lambda \in L} A x_\lambda$, which proves that (c) implies (a). Conversely, suppose that condition (a) holds. Let $A' = A_0[x_\lambda]_{\lambda \in L}$ be the sub-A,-algebra of A generated by the family $(x_\lambda)$ and let us show that $A' = A$. For this, it is sufficient to show that $A_i \subset A$ for all $i \geq 0$. We proceed by induction on $i$, the property being obvious for $i = 0$. Then let $y \in A$, with $i \geq 1$. Since $y \in m$, there exists a family $(a_\lambda)_{\lambda \in L}$ of elements of A of finite support such that $y = \sum_\lambda a_\lambda x_\lambda$ and we may assume that each of the $a_\lambda$ is homogeneous of degree $i - \deg(x_\lambda)$ (by replacing it if need be by its homogeneous component of that degree); as $\deg(x_\lambda) > 0$, the induction hypothesis shows that $a_\lambda \in A'$ for all $\lambda \in L$, whence $y \in A'$ and $A_i \subset A'$, which proves that (a) implies (b).