Category:Graded Submodules
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This category contains results about Graded Submodules.
Definitions specific to this category can be found in Definitions/Graded Submodules.
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 $\forall i : x_i \in M_i$
- $\implies \forall i : x_i \in N$
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