Definition:Graded Module
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Definition
Let $R$ be a graded commutative ring with unity.
The validity of the material on this page is questionable. In particular: It must be specified whether $R$ is $\N$-graded or $\Z$-graded, according to the kind of graded module. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Let $M$ be an $R$-module.
We say $M$ is $\N$-graded if and only if it has a decomposition as a direct sum of abelian groups:
- $\ds R = \bigoplus_{n \mathop \in \N} R_n $
There is believed to be a mistake here, possibly a typo. In particular: This must be $M = \bigoplus_{n \mathop \in \N} M_n$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
such that for all $m, n \in \N$ we have $R_n M_m \subseteq M_{n + m}$.
Similarly, we say $M$ is $\Z$-graded if and only if it has a decomposition as a direct sum of abelian groups:
- $\ds R = \bigoplus_{n \mathop \in \Z} R_n $
There is believed to be a mistake here, possibly a typo. In particular: This must be $M = \bigoplus_{n \mathop \in \Z} M_n$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
such that for all $m, n \in \Z$ we have $R_n M_m \subseteq M_{n + m}$.
Sources
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