Proposition 4

Let A be a graded ring of type \Delta and M a graded A-module of type \Delta. (i) Every sum and every intersection of graded submodules of M is a graded submodule. (ii) If x is a homogeneous element of M of degree \mu which is cancellable in \Delta, the annihilator of x is a graded left ideal of A. (iii) If all the elements of \Delta are cancellable, the annihilator of a graded submodule of M is a graded two-sided ideal of A.

If $(\mathbf{N}_i)$ is a family of graded submodules of $M$, property (c) of Proposition 2 shows that the sum of the $\mathbf{N}_i$ is generated by homogeneous elements and property (b) of Proposition 2 proves that the homogeneous components of every element of $\bigcap_i \mathbf{N}_i$ belongs to $\bigcap_i \mathbf{N}_i$; whence (i).

To prove (ii), it suffices to note that $\operatorname{Ann}(x)$ is the kernel of the homomorphism $a \mapsto ax$ of the $A$-module $A_s$ into $M$ and that this homomorphism is graded of degree $\mu$; the conclusion follows from Proposition 3(ii). Finally (iii) is a consequence of (i) and (ii) for the annihilator of a graded submodule $N$ of $M$ is the intersection of the annihilators of the homogeneous elements of $N$, by virtue of Proposition 2.