# 4.2: Homomorphisms

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##### Learning Objectives

In this section, we'll seek to answer the questions:

• What is a ring homomorphism?
• What are some examples of ring homomorphisms?

Central to modern mathematics is the notion of function 1 . Functions arise in all areas of mathematics, each subdiscipline concerned with certain types of functions. In algebra, our concern is with operation-preserving functions, such as the linear transformations $$L : V\to W$$ of vector spaces you have seen in a course in linear algebra. Those linear transformations had the properties that $$L(\mathbf{v}+\mathbf{u}) = L(\mathbf{v})+L(\mathbf{u})$$ (addition is preserved) and $$L(c\mathbf{u}) = c L(\mathbf{u})$$ (scalar multiplication is preserved).

This section assumes a familiarity with the idea of function from a set-theoretic point of view, as well as the concepts of injective (one-to-one), surjective (onto), and bijective functions (one-to-one correspondences).

We find something similar at work in the study of homomorphisms of rings, which we define to be functions that preserve both addition and multiplication.

##### Definition: Homomorphism

Let $$R$$ and $$S$$ be commutative rings with identity. A function $$\varphi : R\to S$$ is a called ring homomorphism if it preserves addition, multiplication, and sends the identity of $$R$$ to the identity of $$S\text{.}$$ That is, for all $$x,y\in R\text{:}$$

• $$\varphi(x+y) = \varphi(x) + \varphi(y)\text{,}$$
• $$\varphi(xy) = \varphi(x)\varphi(y)\text{,}$$ and
• $$\varphi(1_R) = 1_S\text{.}$$

If $$\varphi$$ is a bijection, we say that $$\varphi$$ is an isomorphism and write $$R\mathbb{C}ong S\text{.}$$ If $$\varphi : R\to R$$ is an isomorphism, we say $$\varphi$$ is an automorphism of $$R\text{.}$$

Our first job when glimpsing a new concept is to collect a stock of examples.

##### Exploration 4.2.1

Determine whether the following functions are homomorphisms, isomorphisms, automorphisms, or none of these. Note that $$R$$ denotes an arbitrary commutative ring with identity.

1. $$\varphi : R\to R$$ defined by $$\varphi(x)=x$$
2. $$\psi : R\to R$$ defined by $$\psi(x)=-x$$
3. $$\alpha : \mathbb{Z}\to \mathbb{Z}$$ defined by $$\alpha(x)=5x$$
4. $$F : \mathbb{Z}_2[x]\to \mathbb{Z}_2[x]$$ defined by $$F(p) = p^2$$
5. $$\iota : \mathbb{C}\to \mathbb{C}$$ defined by $$\iota(a+bi)=a-bi\text{,}$$ where $$a,b\in \mathbb{R}, i^2 = -1$$
6. $$\beta : \mathbb{Z}\to \mathbb{Z}_{5}$$ defined by $$\beta(x) = \overline{x}$$
7. $$\epsilon_r : R[x] \to R$$ defined by $$\epsilon_r(p(x)) = p(r)$$ (this is known as the $$r$$-evaluation map)
8. $$\xi : \mathbb{Z}_5 \to \mathbb{Z}_{10}$$ defined by $$\xi(\overline{x}) = \overline{5x}$$

Homomorphisms give rise to a particularly important class of subsets: kernels.

##### Definition: Kernel

Let $$\varphi : R \to S$$ be a ring homomorphism. Then $$\ker \varphi =\{r\in R : \varphi(r)=0_S\}$$ is the kernel of $$\varphi\text{.}$$

##### Activity 4.2.1

For each homomorphism in Exploration 4.2.1 , find (with justification), the kernel.

In fact, kernels are not just important subsets of rings; they are ideals.

##### Theorem 4.2.1

Given a ring homomorphism $$\varphi : R\to S\text{,}$$ $$\ker\varphi$$ is an ideal

Kernels also give a useful way of determining whether their defining homomorphisms are one-to-one.

##### Theorem 4.2.2

Let $$\varphi : R\to S$$ be a homomorphism. Then $$\varphi$$ is one-to-one if and only if $$\ker\varphi = \{0\}\text{.}$$

This page titled 4.2: Homomorphisms is shared under a not declared license and was authored, remixed, and/or curated by Michael Janssen & Melissa Lindsey via source content that was edited to the style and standards of the LibreTexts platform.