Definition:Graded Submodule

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 $M$.

Definition 1

$N$ is graded if and only if:

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

Definition 2

$N$ is graded if and only if $N$ is generated over $R$ by homogeneous elements of $M$.

Definition 3

$N$ 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 known as

$N$ is also called a homogeneous submodule.

Also see

• Equivalence of Definitions of Graded Submodule
• Definition:Graded Subring

• Results about graded submodules can be found here.