Proposition 4

*Let $A$ be a torsion-free commutative group, $A$ a graded Noetherian ring of type $A$ and $M$ a graded $A$-module of type $A$. Let $p$ be a prime ideal of $A$ and $N$, a submodule of $M$ which is $p$-primary with respect to $M$.

(i) The largest graded ideal $p'$ of $A$ contained in $p$ (Algebra, Chapter 11, § 11, no. 3) is prime.

(ii) The largest graded submodule $N'$ of $N$ is $p'$-primary with respect to $M$.*

We know (loc. cit.) that the homogeneous elements of $p'$ (resp. $N'$) are just the homogeneous elements of $p$ (resp. $N$). Let $a$ be a homogeneous element of $p$; if $x$ is a homogeneous element of $M$, there exists an integer $n > 0$ such that $a^n x \in N$; as $a^n x$ is homogeneous, $a^n x \in N'$; as every $y \in M$ is the direct sum of a finite number of homogeneous elements, we conclude that there exists an integer $q > 0$ such that $a^q y \in N'$, so that the homothety with ratio $a$ in $M/N'$ is almost nilpotent.

Consider now a homogeneous element $b$ of $A - p'$; then $b \notin p$ since $b$ is homogeneous. Let $x$ be an element of $M$ such that $bx \in N'$ and let $(x_i)_{i \in \Delta}$ be the family of homogeneous components of $x$. As $N'$ is graded, $bx_i \in N'$ for all $i$, hence $bx_i \in N$ and, as $b \notin p$, we conclude that $x_i \in N$; as $x_i$ is homogeneous, $x_i \in N'$, whence $x \in N'$ and the homothety with ratio $b$ on $M/N'$ is injective. Proposition 4 then follows from Proposition 3 applied to $p'$ and $M/N'$.