---- > [!definition] Definition. ([[ring homomorphism]]) > Let $R,S$ be [[ring|rings]]. A **ring homomorphism** $\varphi:R \to S$ is a [[group homomorphism|homomorphism]] of the underlying [[abelian group|abelian groups]]: $\varphi(a+b) = \varphi(a) + \varphi(b) \text{ for all } a,b \in R$ that respects multiplication: $\varphi(ab)=\varphi(a)\varphi(b)$ and identity: $\varphi(1_{R})=1_{S}.$ ^definition > [!note] Remark. > In the definition of [[group homomorphism]] there was no need to mandate identity-preservation, for [[group homomorphisms take identities to identities and inverses to inverses|it held automatcially]]. But the proof relied on [[cancellation law for groups|cancellation in groups]]; in particular that if $a=a^{2}$ in a [[group]] $G$ then $e=a$. Of course, in general [[ring|rings]] this fact doesn't hold, e.g., $3=3^{2}$ in $\mathbb{Z}/6\mathbb{Z}$. > > So that proof doesn't carry over; indeed, there exist maps between rings which satisfy all ring homomorphism properties except identity-preservation. Example: $r \mapsto 2r$ on $\mathbb{Z} / 2\mathbb{Z}=(\text{even}, \text{odd})$. $[a] \mapsto [2a]=\text{even}$, $[b] \mapsto [2b]=\text{even}$, $[ab] \mapsto [4ab]=\text{even}$, so multiplication is respected, but $1_{\mathbb{Z} / 2 \mathbb{Z}}=\text{odd}$ maps to $[2]=\text{even}$. > ---- #### ---- #### 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 > ```