Basics of homomorphisms準同型じゅんどうけい

date2026-07-14document_iddoc_3c1f28998382611f90145e3b03a52f40description準同型を、演算を保存する写像として群・環を横断して説明し、核・像・同型への接続を整理する。prerequisites群[準[同型/どうけい]/じゅんどうけい]と[同型/どうけい] / 環の基[本/ほん] / イデアルと[[商/しょう]環/しょうかん] / [整域/せいいき]・[[零/れい]因子/れいいんし]・[[多項[式/しき]/たこうしき]環/たこうしきかん]type講義content_typelecturestatusactive

mathabstract-algebrahomomorphismlecture

A homomorphism準同型じゅんどうけい is a map that preserves operations. In abstract algebra抽象代数ちゅうしょうだいすう, not only the objects themselves but also the maps that move between objects without breaking structure are important.

Just as a linear map線型写像せんけいしゃぞう in linear algebra preserves addition and scalar multiplication, a homomorphism準同型じゅんどうけい in abstract algebra抽象代数ちゅうしょうだいすう preserves the operations of groups or rings.

data/lecture/math/linear-algebra/linear-maps-and-matrices.lecture.n.md

This analogy is only a preview. The definitions and proofs on this page use only groups, rings, and maps introduced earlier.

1Group homomorphisms準同型じゅんどうけい

A map $φ : G \to H$ between groups $G, H$ is a group homomorphism群準同型ぐんじゅんどうけい if, for all $a, b \in G$ , it satisfies

φ (a b) = φ (a) φ (b)

Identity elements and inverses follow from this condition. Indeed,

φ (e_{G}) = φ (e_{G} e_{G}) = φ (e_{G})^{2},

and multiplying by $φ (e_{G})^{- 1}$ gives $φ (e_{G}) = e_{H}$ . Moreover,

e_{H} = φ (e_{G}) = φ (a a^{- 1}) = φ (a) φ (a^{- 1}),

so $φ (a^{- 1}) = φ (a)^{- 1}$ .

2Ring homomorphisms準同型じゅんどうけい

A map $φ : R \to S$ between rings $R, S$ is a ring homomorphism環準同型かんじゅんどうけい if, for all $a, b \in R$ , it satisfies

φ (a + b) = φ (a) + φ (b)

φ (a b) = φ (a) φ (b)

and also

φ (1_{R}) = 1_{S}

In this material, rings have identity elements, and ring homomorphisms preserve them by default.

The additive condition also implies $φ (0_{R}) = 0_{S}$ and $φ (- a) = - φ (a)$ . Indeed,

φ (0_{R}) = φ (0_{R} + 0_{R}) = φ (0_{R}) + φ (0_{R})

gives $φ (0_{R}) = 0_{S}$ , while

0_{S} = φ (0_{R}) = φ (a + (- a)) = φ (a) + φ (- a)

gives $φ (- a) = - φ (a)$ .

3Kernel and image像ぞう

The kernel核かく of a homomorphism準同型じゅんどうけい is the part collapsed to the identity element単位元たんいげん or to 0.

For a group homomorphism群準同型ぐんじゅんどうけい,

\ker φ = {g \in G ∣ φ (g) = e_{H}}

For a ring homomorphism環準同型かんじゅんどうけい,

\ker φ = {r \in R ∣ φ (r) = 0_{S}}

For a map $φ : A \to B$ , the image像ぞう is the set of all elements of $B$ actually reached:

Im φ = {φ (x) ∣ x \in A} \subseteq B .

4What changes and what is preserved

In a homomorphism準同型じゅんどうけい, the names or representations of elements may change. However, operating first and then mapping gives the same result as mapping first and then operating. In this sense, a homomorphism準同型じゅんどうけい is a structure-preserving map.

An isomorphism同型どうけい is an invertible homomorphism準同型じゅんどうけい. When an isomorphism同型どうけい exists, the structure is completely preserved.

In particular, a bijective ring homomorphism is called a ring isomorphism環同型かんどうけい. Its inverse also preserves addition, multiplication, and the identity. Indeed, writing $y = φ (a)$ and $z = φ (b)$ gives

φ^{- 1} (y + z) = φ^{- 1} (φ (a + b)) = a + b = φ^{- 1} (y) + φ^{- 1} (z),

φ^{- 1} (y z) = φ^{- 1} (φ (a b)) = a b = φ^{- 1} (y) φ^{- 1} (z),

and $φ^{- 1} (1_{S}) = φ^{- 1} (φ (1_{R})) = 1_{R}$ .

5Example: evaluation map評価写像ひょうかしゃぞうs are ring homomorphisms環準同型かんじゅんどうけい

For a field体たい $F$ and $c \in F$ , the map from the polynomial ring多項式環たこうしきかん $F [x]$ to $F$ defined by

{ev}_{c} : F [x] \to F, f (x) \mapsto f (c)

is called the evaluation map評価写像ひょうかしゃぞう. It is a ring homomorphism環準同型かんじゅんどうけい because

{ev}_{c} (f + g) = f (c) + g (c)

and

{ev}_{c} (f g) = f (c) g (c)

hold, and ${ev}_{c} (1) = 1$ .

The kernel核かく of this map is the set of all polynomials that have $c$ as a root:

\ker ({ev}_{c}) = {f (x) \in F [x] ∣ f (c) = 0}

The fact that this kernel核かく is an idealイデアル is a basic example connecting homomorphisms準同型じゅんどうけい with quotient rings商環しょうかん.

6Proof supplement: the kernel核かく of a ring homomorphism環準同型かんじゅんどうけい is an idealイデアル

Let $φ : R \to S$ be a ring homomorphism環準同型かんじゅんどうけい. Define its kernel核かく by

\ker φ = {r \in R ∣ φ (r) = 0} .

First, $φ (0_{R}) = 0_{S}$ , so $0_{R} \in \ker φ$ ; hence the kernel is nonempty. If $a, b \in \ker φ$ , then

φ (a - b) = φ (a) - φ (b) = 0 - 0 = 0,

so $a - b \in \ker φ$ . Also, if $r \in R$ and $a \in \ker φ$ , then

φ (r a) = φ (r) φ (a) = φ (r) 0 = 0, φ (a r) = φ (a) φ (r) = 0 φ (r) = 0,

so $r a, a r \in \ker φ$ . Therefore the kernel核かく of a ring homomorphism環準同型かんじゅんどうけい is an idealイデアル.

We also verify closure of the image像ぞう. Let $x = φ (a)$ and $y = φ (b)$ lie in $Im φ$ . Then

x + y = φ (a + b), - x = φ (- a), x y = φ (a b),

and both $0_{S} = φ (0_{R})$ and $1_{S} = φ (1_{R})$ lie in the image. Thus the image is closed under addition, additive inverses, and multiplication and contains the zero and multiplicative identity. Associativity of addition and multiplication, commutativity of addition, and the distributive laws are inherited by restricting the operations of $S$ . Here, a subset that is a ring under the operations inherited from $S$ and shares the multiplicative identity of $S$ is called a subring of $S$ . Therefore $Im φ$ is a subring of $S$ .

7Exercise link

data/exercise/math/abstract-algebra/homomorphisms-and-isomorphisms.exercise.n.md

8Summary

A homomorphism準同型じゅんどうけい is a map that preserves operations. A group homomorphism preserves the group product, while a ring homomorphism preserves addition, multiplication, and the multiplicative identity. The kernel核かく is the part mapped to an identity element or to 0, and the image像ぞう is the part actually reached. Evaluation maps provide concrete polynomial examples, and the kernel of a ring homomorphism is an ideal. These ideas lead to quotient structures and the first isomorphism theorem.