Definition:Graded Submodule/Definition 1

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:

$\ds N = \bigoplus_{n \mathop \in G} \paren {N \cap M_n}$

Also see

• Equivalence of Definitions of Graded Submodule

• Results about graded submodules can be found here.

Sources

• 1980: Hideyuki Matsumura: Commutative Algebra $10:$ Graded Ring and Modules