4.2: Homomorphisms

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 function¹. 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 \(\ker\varphi = \{0\}\text{.}\)