----- $R$ is a [[ring]]. > [!proposition] Proposition. ([[cancellation characterization of zero division]]) > $a \in R$ is *not* a left- (resp., right-) [[zero-divisor]] if and only if left- (resp., right) multiplication is an [[injection|injective]] function $R \to R$, i.e., iff $\underbrace{ar_{1}=ar_{2} \implies r_{1}=r_{2}}_{\text{left}} \text{ resp. } \underbrace{r_{1}a=r_{2}a \implies r_{1}=r_{2}}_{\text{right}}.$ ^proposition > [!proof]- Proof. ([[cancellation characterization of zero division]]) > **Left.** Assume $a$ is not a left-[[zero-divisor]] and $ab=ac$ for some $b,c \in R$. Then by distributivity and the existence of additive inverses, $a(b-c)=0$, hence $b-c=0$ so that $b=c$. > Conversely, if $a$ is a left-[[zero-divisor]], then there exists $b \in R$ such that $ab=0$. Then $f(b)=0=f(0)$ and so $f$ must not be an [[injection]]. > **Right.** This case is totally analogous. ----- #### ---- #### 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 > ```