Ring Homomorphisms and Ideals

Ring homomorphisms are the structure-preserving maps between rings. Their kernels are ideals, and the interplay between homomorphisms and ideals is central to the structure theory of rings.

Ring Homomorphisms

Definition5.1Ring homomorphism

A ring homomorphism $\varphi: R \to S$ between rings (with unity) is a function satisfying:

$\varphi(a + b) = \varphi(a) + \varphi(b)$ for all $a, b \in R$ .
$\varphi(ab) = \varphi(a)\varphi(b)$ for all $a, b \in R$ .
$\varphi(1_R) = 1_S$ .

A ring isomorphism is a bijective ring homomorphism. An endomorphism is a homomorphism from a ring to itself; an automorphism is a bijective endomorphism.

ExampleRing homomorphisms

The inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q} \hookrightarrow \mathbb{R} \hookrightarrow \mathbb{C}$ .
The reduction map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ , $a \mapsto a \bmod n$ .
The evaluation map $R[x] \to R$ , $f(x) \mapsto f(a)$ for fixed $a \in R$ .
The Frobenius endomorphism $\varphi: \mathbb{F}_p[x] \to \mathbb{F}_p[x]$ , $f(x) \mapsto f(x^p)$ .
Complex conjugation $\mathbb{C} \to \mathbb{C}$ , $z \mapsto \bar{z}$ is a ring automorphism.

The Correspondence Theorem

Theorem5.1Ideal correspondence theorem

Let $\varphi: R \to S$ be a surjective ring homomorphism with kernel $I = \ker(\varphi)$ . There is a bijection:

$\{\text{ideals } J \text{ of } R \text{ with } I \subseteq J\} \longleftrightarrow \{\text{ideals } K \text{ of } S\}$

given by $J \mapsto \varphi(J)$ and $K \mapsto \varphi^{-1}(K)$ . This bijection preserves:

Inclusion: $J_1 \subseteq J_2 \iff \varphi(J_1) \subseteq \varphi(J_2)$ .
Sums: $\varphi(J_1 + J_2) = \varphi(J_1) + \varphi(J_2)$ .
Intersections: $\varphi(J_1 \cap J_2) = \varphi(J_1) \cap \varphi(J_2)$ .
Prime/maximal ideals: $J$ is prime (resp. maximal) iff $\varphi(J)$ is.

ExampleIdeals of quotient rings

The ideals of $\mathbb{Z}/12\mathbb{Z}$ correspond to ideals of $\mathbb{Z}$ containing $(12)$ : $(1), (2), (3), (4), (6), (12)$ . So $\mathbb{Z}/12\mathbb{Z}$ has ideals $\{0\}, (\bar{2}), (\bar{3}), (\bar{4}), (\bar{6}), \mathbb{Z}/12\mathbb{Z}$ .

Operations on Ideals

RemarkIdeal arithmetic

For ideals $I, J$ of a ring $R$ :

Sum: $I + J = \{a + b : a \in I, b \in J\}$ (smallest ideal containing both).
Product: $IJ = \{\sum a_i b_i : a_i \in I, b_i \in J\}$ (generated by pairwise products).
Intersection: $I \cap J$ is always an ideal.
Radical: $\sqrt{I} = \{r \in R : r^n \in I \text{ for some } n \geq 1\}$ .

Key relations: $IJ \subseteq I \cap J \subseteq I, J \subseteq I + J$ , and $I + J = R$ iff $I$ and $J$ are comaximal.