Proposition 1

Let $A$ be a graded commutative ring of type $\Delta$, $E$ a graded $A$-algebra of type $\Delta$ and $S$ a set of homogeneous elements of $E$. Then the sub-$A$-algebra (resp. left ideal, right ideal, two-sided ideal) generated by $S$ is a graded subalgebra (resp. graded ideal)..

The subalgebra of $E$ generated by $S$ is the sub-$A$-module generated by the finite products of elements of $S$, which are homogeneous; similarly, the left (resp. right) ideal generated by $S$ is the sub-$A$-module generated by the elements of the form $u_1(u_2(\ldots(u_n s))\ldots)$ (resp. $(\ldots((s u_n) u_{n-1})\ldots) u_2$ ) $u_1$) where s ∈ S and the u_j ∈ E are homogeneous (n arbitrary) and these products are homogeneous, whence in this case the conclusion by virtue of II, § 11, no. 3, Proposition 2); finally the two-sided ideal generated by S is the union of the sequence (\mathfrak{I}_n)_{n \geq 1}, where \mathfrak{I}_1 is the left ideal generated by S and \mathfrak{I}_{2n} (resp. \mathfrak{I}_{2n+1}) the right (resp. left) ideal generated by \mathfrak{I}_{2n-1} (resp. \mathfrak{I}_{2n}), which completes the proof.