---- > [!definition] Definition. ([[integral domain]]) > A **integral domain** is a nonzero [[commutative ring]] $R$ (with $1$) such that for all $a,b \in R$, $ab=0 \implies a=0 \text{ or } b=0,$ > i.e., a [[ring]] without [[zero-divisor|zero-divisors]]. > ^definition > [!basicproperties] > - By [[cancellation characterization of zero division]], cancellation holds in integral domains ^properties > [!basicexample] > - $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ are all integral domains. > - More generally, any [[field]] $\mathbb{F}$ is an [[integral domain]]. For if $ab=0$ with (WLOG) $b \neq 0$, then $ab b^{-1}=0b ^{-1}=0$ hence $a=0$. ^basic-example > [!basicnonexample] > $\mathbb{Z} / n\mathbb{Z}$ is *not* an integral domain unless $n$ is [[prime number|prime]]. For $a | n$ for with $a \neq 1$ and $a \neq n$ and [[divides|divisor]] $q$, we have $aq=n=0$. That said, prime $n$ suffices to turn $\mathbb{Z} / n\mathbb{Z}$ into an integral domain, because if $n$ is prime then by [[finite subgroup of multiplicative field of a group is cyclic]] we know that multiplication on $\mathbb{Z} / n\mathbb{Z}$ turns it into a [[cyclic group]], and [[cancellation law for groups|cancellation]] holds in [[group|groups]]. ^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 > ```