---- > [!definition] Definition. ([[kernel of a ring homomorphism]]) > The **kernel** of a [[group homomorphism|homomorphism]] $\varphi:R \to S$ of [[ring|rings]] is $\text{ker }\varphi:=\{ r \in R : \varphi(r)=0 \},$ > that is, the [[kernel of a group homomorphism|kernel]] of $\varphi$ when viewed as a [[group homomorphism]]. > > Note that this is *not* a kernel [[categorical kernel|in the category-theoretic]] sense — indeed, $\mathsf{Ring}$ does not have a [[terminal object|zero object]]. ^definition > [!basicnonexample] Warning. > Unlike with [[group|groups]], the [[kernel of a ring homomorphism]] is not in general a [[subring]]. For [[subring|subrings]] must contain multiplicative identity, hence if $\ker \varphi$ is to be a [[subring]] we must have $\varphi(1_{R})=1_{S}=0_{S}$, which only happens when $S$ is the zero-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 > ```