---- > [!definition] Definition. ([[subring]]) > A **subring** of a [[ring]] $R$ is a subset $S \subset R$ for which > - $S$ is a [[subgroup]] of the [[abelian group]] $(R,+)$; > - $S$ is closed under multiplication; > - $1_{R} \in S$. ^definition > [!basicnonexample] > A departure from the case with [[group|groups]] is that the zero-ring $(0)$ is not a subring of any nonzero [[ring]]. ^nonexample ---- #### ---- #### 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 > ```