Guiding Questions.
In this section, we’ll seek to answer the questions:
- What is a ring homomorphism?
- What are some examples of ring homomorphisms?
Section 4.2 Homomorphisms
Central to modern mathematics is the notion of function. 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).
1
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 4.2.1.
Let \(R\) and \(S\) be commutative rings with identity. A function \(\p : R\to S\) 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 \(x,y\in R\text{:}\)
- \(\p(x+y) = \p(x) + \p(y)\text{,}\)
- \(\p(xy) = \p(x)\p(y)\text{,}\) and
- \(\p(1_R) = 1_S\text{.}\)
If \(\p\) is a bijection, we say that \(\p\) is an isomorphism and write \(R\cong S\text{.}\) If \(\p : R\to R\) is an isomorphism, we say \(\p\) 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 and \(F\) a field.
- \(\p : R\to R\) defined by \(\p(x)=x\)
- \(\psi : R\to R\) defined by \(\psi(x)=-x\)
- \(\alpha : \Z\to \Z\) defined by \(\alpha(x)=5x\)
- \(F : \Z_2[x]\to \Z_2[x]\) defined by \(F(p) = p^2\)
- \(\iota : \C\to \C\) defined by \(\iota(a+bi)=a-bi\text{,}\) where \(a,b\in \R, i^2 = -1\)
- \(\beta : \Z\to \Z_{5}\) defined by \(\beta(x) = \overline{x}\)
- \(\epsilon_r : F[x] \to F\) defined by \(\epsilon_r(p(x)) = p(r)\text{,}\) where if \(p(x) = a_0 + a_1 x + \cdots + a_n x^n\text{,}\) \(p(r)\) is the expression obtained by “plugging \(r\) into \(p\)”: \(p(r) = a_0 + a_1 r + \cdots + a_n r^n\) (this is known as the \(r\)-evaluation map)
- \(\xi : \Z_5 \to \Z_{10}\) defined by \(\xi(\overline{x}) = \overline{5x}\)
Homomorphisms give rise to a particularly important class of subsets: kernels.
Definition 4.2.2.
Let \(\p : R \to S\) be a ring homomorphism. Then \(\ker \p =\setof{r\in R}{\p(r)=0_S}\) is the kernel of \(\p\text{.}\)
Activity 4.2.2.
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 4.2.3.
Given a ring homomorphism \(\p : R\to S\text{,}\) \(\ker\p\) is an ideal.
Kernels also give a useful way of determining whether their defining homomorphisms are one-to-one.
Theorem 4.2.4.
Let \(\p : R\to S\) be a homomorphism. Then \(\p\) is one-to-one if and only if \(\ker\p = \set{0}\text{.}\)
