# Ring Homomorphisms

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:

(1)

\begin{align} \quad \phi (r) = r(x) \end{align}

Where $r(x) = r$, i.e., $r(x)$ is the constant polynomial that gives the value $r$.

Let $a, b \in R$. Then:

(2)

\begin{align} \quad \phi (a + b) = (a + b)(x) = a(x) + b(x) = \phi (a) + \phi(b) \end{align}

(3)

\begin{align} \quad \phi (a * b) = (a * b)(x) = a(x) * b(x) = \phi(a) * \phi(b) \end{align}

Furthermore, if $1 \in R$ is the multiplicative identity in $R$ then $1(x)$ is the multiplicative identity in $R[x]$ and:

(4)

\begin{align} \quad \phi(1) = 1(x) \end{align}

So $\phi$ is a homomorphism from $R$ to $R[x]$. So $R$ and $R[x]$ are homomorphic.

### Related post:

- Photos: Warwick Academy Mathematics Day – Bernews
- Top Scientists Honored at 8th Annual Breakthrough Prize Ceremony in Silicon Valley – EurekAlert
- I am not marrying June Malia, says Sourav Chatterjee – The Asian Age
- Second week of HSC wraps up, Bathurst students complete Mathematics exams – Western Advocate
- AI and mathematics – Times of Malta
- UPPSC LT Grade Result 2018 for Mathematics Released at uppsc.up.nic.in, Direct Link – News18
- CBSE 10th Maths sample paper 2020, exam pattern & important tips: Mathematics Basic & Mathematics Standar – Times of India
- Matric Exams: Technical Mathematics Paper 1 past paper 2018 – IOL
- Explained: Why is there no mathematics Nobel? The theories, the facts, the myths – The Indian Express
- CBSE New Circular All About ‘Mathematics’ Challenge – Odisha Television Ltd.