---- > [!definition] Definition. ([[graded module]]) > Let $A=\bigoplus_{n \geq 0}A_{n}$ be a [[graded ring]]. A **graded $A$-module** is an $A$-[[module]] $M$ admitting [[direct sum of modules|direct sum decomposition]] $M=\bigoplus_{n \geq 0}M_{n},$ > where $A_{m}M_{n} \subset M_{m+n}$ (in particular, $M_{n}$ is an $A_0$-[[module]].) Nonzero elements of $M_{i}$ are called **homogeneous of degree $i$**. > > A **graded $A$-module morphism** is a [[linear map|homomorphism]] $f:\bigoplus_{\ell \geq 0}M_{\ell} \to \bigoplus_{\ell \geq 0}N_{\ell}$ of graded $A$-modules that preserves grading: $f(M_{\ell}) \subset N_{\ell}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```