Remark 2

Let $M$ be a graded $A$-module and $E$ a submodule of $M$; it follows from Proposition 4(i) that there exists a largest graded submodule $N'$ of $M$ contained in $E$ and a smallest graded submodule $N''$ of $M$ containing $E$; $N'$ is the set of $x \in E$ all of whose homogeneous components belong to $E$ and $N''$ is the submodule of $M$ generated by the homogeneous components of a generating system of $E$.