----- > [!proposition] Proposition. ([[every nonzero ring homomorphism out of a field is injective]]) > Let $\varphi: k \to R$ be a [[ring homomorphism]], where $k$ is a [[field]] and $R$ is a nonzero [[ring]]. Then $\varphi$ is an [[injection]]. ^proposition > [!proof]- Proof. ([[every nonzero ring homomorphism out of a field is injective]]) > Recall [[kernel iff ideal]] and [[division ring iff ideals are {0} and R|commutative ring is a field iff ideals are {0} and R]]. Thus the [[kernel of a ring homomorphism|kernel]] of $\varphi$ is either $\{ 0 \}$ or $k$. But if it is $k$ then the [[first isomorphism theorem for rings]] implies $k / k \cong \{ 0 \}$ is [[ring isomorphism|isomorphic]] to $R$, which can't happen because we assumed $R \neq \{ 0 \}$. Thus $\ker \varphi=\{ 0 \}$ and we are done by [[ring homomorphism is injective iff kernel is trivial iff is a monomorphism]]. ----- #### ---- #### 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 > ```