Homomorphism

Definition: A homomorphism is a “Structure-Preserving Map”

Group Homomorphism:

$f:G\rightarrow H \\ f(g_1\cdot g_2)=f(g_1)\cdot f(g_2)$

(It follows that $f(e_G)=e_H$, identity mapped to identity)

Example:

$f:\mathbb{Z} \rightarrow \mathbb{Z}_n$

Addition group of integers → $(\{0,1,\cdots, n-1, +_{mod\ n}\})$

$f(a) = a_{(mod\ n)}$

Definition: A function $f:R\rightarrow S$ (where $R$ and $S$ are rings) is a ring homomorphism if:

1. $f(r_1+_Rr_2)=f(r_1)+_Sf(r_2) \ \forall\ r_1,r_2 \in R$ (Namely, $f$ is a group homomorphism $f:(R,+)\rightarrow(s,+)$)
2. $f(r_1\cdot_Rr_2)=f(r_1)\cdot_Sf(r_2)\ \forall\ r_1,r_2\in R$
3. $f(1_R)=1_S$

Remark: It is guarenteed that if $f:R\rightarrow S$ is a ring homomoprhism, then $f(0_R) =0_S$

Examples:

1. For any ring R, the identity map $id_R: R\rightarrow R\\ \ \ \ \ \ \ \ \ \ \ r\mapsto r$ is a ring homomorphism
2. If $S\subseteq R$ is a subring, then the inclusion map: $i_S:S\rightarrow R \\ \ \ \ \ \ \ \ s\mapsto s$ is a ring homomorphism. E.g., $f:Q\rightarrow R \\ \ f(\alpha)\mapsto \alpha$
3. $f: \mathbb{Z} \rightarrow \mathbb{Z}n = \{0, 1, \cdots, n-1\}\\ f(a) = a{(mod\ n)}$

• $f(a+b)=f(a)+f(b)$
• $f(a\cdot b) = f(a) \cdot f(b):\\ \ (a\cdot b){mod\ n} = a{mod\ n}\cdot b_{mod\ n}$