---- > [!definition] Definition. ([[homogeneous ideal]]) > An [[ideal]] $I$ of a [[graded ring]] $S$ is said to be **homogeneous** if it is [[ideal generated by a subset|generated by]] its homogeneous elements, i.e., if it is [[ideal generated by a subset|generated by]] by the subset $\bigcup_{d}I \cap S_{d}.$ ^definition > [!basicexample] > $\langle x \rangle$, $\langle x, y \rangle$, $\langle x^{2}+y^{2} \rangle$, $\langle x^{2}, xy \rangle$ are all homogeneous ideals, by their construction. ^basic-example > [!basicproperties] > [[prime ideal|Primality]] of a homogeneous ideal $I$ can be checked on its homogeneous elements. That is, for any $s \in S_{n}$ and $t \in S_{m}$, $st \in I$ implies $s \in I$ or $t \in I$. ^properties ---- #### ---- #### 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 > ```