Basics of rings環かん

date2026-07-14document_iddoc_b5d7f2cff0a5b08888fa59db9166010edescription環を、加法の可換群構造と乗法のモノイド構造を分配法則で結んだものとして定義し、基本恒等式と整数・多項式・行列・剰余類の例を説明する。prerequisites群の基[本/ほん] / [半群/はんぐん]・モノイド・[群/ぐん]への[入口/いりぐち] / [合[同/どう][式/しき]/ごうどうしき]とmod[演算/えんざん]の基[本/ほん]type講義content_typelecturestatusactive

mathabstract-algebraring-theorylecture

A group群ぐん was a structure with one operation演算えんざん. A ring環かん is a structure構造こうぞう with two operations at the same time: addition加法かほう and multiplication乗法じょうほう.

The reason to study rings is to handle integers整数せいすう, polynomials多項式たこうしき, matrices行列ぎょうれつ, and residue classes剰余類じょうよるい in the same language. They look different, but their addition and multiplication are connected by distributive laws分配法則ぶんぱいほうそく.

1Definition of a ring

A triple $(R, +, \cdot)$ consisting of a set集合しゅうごう $R$ and two operations演算えんざん $+$ and $\cdot$ is a ring環かん if it satisfies the following conditions.

First, $(R, +)$ is an abelian group可換群かかんぐん. Thus addition is associative and commutative, has an additive identity 0, and each element元げん $a$ has an additive inverse加法逆元かほうぎゃくげん $- a$ .

Next, $(R, \cdot)$ is a monoid. Thus multiplication satisfies the associative law結合法則けつごうほうそく and has a multiplicative identity乗法単位元じょうほうたんいげん $1$ .

(a b) c = a (b c)

1 a = a 1 = a

Furthermore, addition and multiplication are connected by the distributive laws分配法則ぶんぱいほうそく.

a (b + c) = a b + a c

(a + b) c = a c + b c

These axioms hold for every choice of $a, b, c \in R$ . Some conventions do not require $1$ ; in this material, unless stated otherwise, rings have a multiplicative identity. This definition includes the zero ring, in which $0 = 1$ . Structures that exclude the zero ring, such as fields and integral domains, explicitly require $0 \neq 1$ in their definitions.

2Why multiplicative inverses are not required

In the integer ring $Z$ , the multiplicative inverse of 2 does not exist among the integers. If the definition定義ていぎ of a ring required every nonzero element to have a multiplicative inverse, the integers could not be treated as a ring.

A ring does not require division. Instead, it preserves the structure in which addition and multiplication work together.

A commutative ring with $0 \neq 1$ in which every nonzero element has a multiplicative inverse is a field体たい.

data/lecture/math/abstract-algebra/field-basics.lecture.n.md

3Basic examples

ring環かん	Addition	Multiplication	Feature
$Z$	Addition of integers	Multiplication of integers	Division is generally not closed
$Z / n Z$	Addition of residue classes	Multiplication of residue classes	For composite moduli, two nonzero classes can have product zero
$Z [x]$	Addition of polynomials	Multiplication of polynomials	Polynomial ring with integer coefficients
$M_{n} (Z)$	Addition of matrices	Matrix multiplication	Noncommutative when $n [PARSE ERROR: Undefined("Command(\"ge\")")] 2$

Because matrix multiplication is noncommutative, commutativity of multiplication is not always required in rings.

data/lecture/math/linear-algebra/meaning-of-matrix-multiplication.lecture.n.md

The example $M_{n} (Z)$ for $n [PARSE ERROR: Undefined("Command(\"ge\")")] 2$ is included as a standard noncommutative ring. The detailed theory of matrices is not needed for the definition of rings in this chapter.

4What is preserved

In a ring環かん, an additive abelian-group structure, a multiplicative monoid structure, and the distributive laws hold simultaneously. Looking only at addition gives an abelian group; looking only at multiplication gives a monoid. The distributive laws connect the two, making calculations such as expansion and factorization possible.

5Basic identities derived from the axioms

For every $a, b \in R$ in a ring,

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

hold. These are not additional axioms; they follow from the distributive laws and additive inverses.

For example,

a 0 = a (0 + 0) = a 0 + a 0.

Adding $- (a 0)$ to both sides in the additive group gives $0 = a 0$ . Moreover,

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

so $(- a) b$ is the additive inverse of $a b$ and hence $(- a) b = - (a b)$ . The proof of $a (- b) = - (a b)$ is analogous. Consequently,

(- a) (- b) = a b

also follows, since $(- a) (- b) = - ((- a) b) = - (- (a b)) = a b$ .

6Exercise link

data/exercise/math/abstract-algebra/rings-ideals-and-quotient-rings.exercise.n.md

7Supplement: units and the difference from fields体たい

An element $u$ of a ring環かん $R$ is a unit単元たんげん if there exists $v \in R$ such that

u v = v u = 1

In a field体たい, every nonzero element is a unit. In a general ring, however, only some elements are units.

For example, the only units in $Z$ are $1$ and $- 1$ . The element 2 is an element of $Z$ , but there is no integer $v$ such that $2 v = 1$ . Thus in rings, being able to multiply and being able to divide must be distinguished.

8Warning: noncommutative rings

The matrix ring $M_{n} (Z)$ is not commutative when $n [PARSE ERROR: Undefined("Command(\"ge\")")] 2$ : there exist matrices $A, B$ for which $A B \neq B A$ . Therefore, in ring calculations, the order順序じゅんじょ of multiplication cannot be changed freely. When an argument uses commutativity, the assumption that the ring is commutative must be stated explicitly.

9Summary

A ring環かん is an algebraic structure代数的構造だいすうてきこうぞう that connects an additive abelian group and a multiplicative monoid by distributive laws. It allows integers, polynomials, residue classes, and matrices to be treated in a common language. Multiplicative inverses are not required for every element; the elements that do have inverses are called units. Next, we study ideals, the subsets used to construct quotient rings.