Proposition 2

Let A be a graded ring of type $\Delta$, M a graded A-module of type $\Delta$, $(M_\lambda)$ its graduation and N a sub-A-module of M. The following properties are equivalent:

(a) $N$ is the sum of the family $(N \cap M_\lambda)_{\lambda \in \Delta}$. (b) The homogeneous components of every element of N belong to N. (c) $N$ is generated by homogeneous elements.

Every element of N can be written uniquely as a sum of elements of the $M_\lambda$ and hence it is immediate that (a) and (b) are equivalent and that (a) implies (c). We show that (c) implies (b). Then let $(x_i)_{i \in I}$ be a family of homogeneous generators $\neq 0$ of N and let $\delta(i)$ be the degree of $x_i$. Every element of N can be written as $\sum_i a_i x_i$ with $a_i \in A$; if $a_{i,\lambda}$ is the component of $a_i$ of degree $\lambda$, the conclusion follows from the relation

Remark (1) In the above notation, the relation $\sum_{i \in I} a_i x_i = 0$ is therefore equivalent to the system of relations $\sum_{\mu + \delta(i) = \lambda} a_{i,\mu} x_i = 0$. When $\Delta$ is a group, these relations can be written $\sum_{i \in I} a_{i,\lambda - \delta(i)} x_i = 0$.

When a submodule N of M has the equivalent properties stated in Proposition 2, clearly the $N \cap M_\lambda$ form a graduation compatible with the A-module structure of N, called the graduation induced by that on M; N with this graduation is called a graded submodule of M.