We have already defined Group Homomorphisms. We now extend the concept to rings.
|Definition: Let $(R, +_1, *_1)$ and $(S, +_2, *_2)$ be rings with multiplicative identities $1_R$ and $1_S$ respectively. A function $\phi : R \to S$ is a Ring Homomorphism if:
1) $\phi(a +_1 b) = \phi(a) +_2 \phi(b)$ for all $a, b \in R$.
2) $\phi(a *_1 b) = \phi(a) *_2 \phi(b)$ for all $a, b \in R$.
3) $\phi(1_R) = 1_S$.
If such a ring homomorphism exists then $(R, +_1, *_1)$ and $(S, +_2, *_2)$ are said to be Homomorphic.
For example, consider the ring $(R, +, *)$ and the polynomial ring $(R[x], +, *)$. We will show that $R$ is homomorphic to $R[x]$. Let $\phi : R \to R[x]$ be defined for all $r \in R$ by:
Where $r(x) = r$, i.e., $r(x)$ is the constant polynomial that gives the value $r$.
Let $a, b \in R$. Then:
Furthermore, if $1 \in R$ is the multiplicative identity in $R$ then $1(x)$ is the multiplicative identity in $R[x]$ and:
So $\phi$ is a homomorphism from $R$ to $R[x]$. So $R$ and $R[x]$ are homomorphic.
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