---- > [!definition] Definition. ([[irrelevant ideal]]) > Let $S$ be a [[graded ring]]: $S=\bigoplus_{d \geq 0} S_{d}$ with $S_{d} \cdot S_{e} \subset S_{d+e}$. We call the [[ideal]] of positive-degree elements $S_{+}:= \bigoplus_{ d \geq 1}S_{d}$ of $S$ the **irrelevant ideal**. Note that $S_{+}=\ker (S \twoheadrightarrow S_{0})$, [[first isomorphism theorem for modules|hence]] $\frac{S}{S_{+}} \cong S_{0}$. ^definition > [!basicexample] > In the [[polynomial 4|polynomial]] [[ring]] $k[X_{0},\dots,X_{n}]$ with the natural grading, the irrelevant ideal is $\langle X_{0},\dots,X_{n} \rangle$. ^basic-example ---- #### ---- #### 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 > ```