---- > [!definition] Definition. ([[characteristic of a ring]]) > [[ring#^properties|Recall that]] $(\mathbb{Z}, +, \cdot)$ is [[terminal object|initial]] in the [[category]] $\mathsf{Ring}$; let $f:\mathbb{Z} \to R$ be the unique [[ring homomorphism]], defined by $a \mapsto \overbrace{1_{R} + \dots + 1_{R}}^{a \text{ times}}$. Then the [[kernel of a ring homomorphism|kernel]] of $f$ equals $n\mathbb{Z}$ for some $n \in \mathbb{Z}_{\geq 0}$. We call $n$ the **characteristic** of the [[ring]] $R$: it is the order of the multiplicative identity $1_{R}$ as an element of the [[group]] $(R,+)$ if $n>0$, or $0$ if $1_{R}$ has additive [[order of an element in a group|order]] $\infty$. ^definition > [!basicexample] > - The characteristic of a [[field]] $k$ is either $0$ or a prime number $p$. Indeed, since [[field|fields]] are [[integral domain|integral domains]], the image $f(\mathbb{Z})$ must be an [[integral domain]] (as a [[subring]] of an [[integral domain]]). By the relevant first isomorphism theorem, $f(\mathbb{Z}) \cong \frac{\mathbb{Z}}{\ker f}$, hence $\frac{\mathbb{Z}}{\ker f}$ must be an [[integral domain]], which only happens if $\ker f=p\mathbb{Z}$ for prime $p$ or is otherwise trivial. ^basic-example ---- #### ---- #### 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 > ```