Definition:Graded Submodule/Definition 3
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Definition
Let $G \in \set {\N, \Z}$.
Let $R$ be a $G$-graded commutative ring with unity.
Let $\ds M = \bigoplus_{n \mathop \in G} M_n$ be a $G$-graded $R$-module.
Let $N$ be a submodule of $M$.
$N$ is graded if and only if:
- $x_r + x_{r+1} + \cdots + x_s \in N$ such that $\forall i : x_i \in M_i$
- $\implies \forall i : x_i \in N$
Also see
- Results about graded submodules can be found here.
Sources
- 1980: Hideyuki Matsumura: Commutative Algebra $10:$ Graded Ring and Modules