环²

环

定义: A ring $R$ is an abelian group with a multiplication operation $(a, b) \rightarrow a b$ that is associative and satisfies the distributive laws: $a(b+c)=a b+a c$ and $(a+b) c=a b+a c$ for all $a, b, c \in R$.

We will always assume that $R$ has at least two elements, including a multiplicative identity $1_R$ satisfying $a 1_R=1_R a=a$ for all $a$ in $R$. The multiplicative identity is often written simply as $1$, and the additive identity as $0$.

定义: If $a$ and $b$ are nonzero but $ab=0$, we say that $a$ and $b$ are zero divisors;

if $a\in R$ and for some $b\in R$ we have $ab=ba=1$, we say that $a$ is a unit or that $a$ is invertible.

交换环, 整环, 除环, 域, 特征

定义: Note that $a b$ need not equal $b a$; if this holds for all $a, b \in R$, we say that $R$ is a commutative ring.

An integral domain is a commutative ring with no zero divisors.

A division ring or skew field is a ring in which every nonzero element $a$ has a multiplicative inverse $a^{-1}$, (i.e., $a a^{-1}=a^{-1} a=1$). Thus the nonzero elements form a group under multiplication.

A field is a commutative division ring.

The characteristic of a ring $R$ (written $\mathrm{Char}R$) is the smallest positive integer such that $n 1=0$, where $n 1$ is an abbreviation for $1+1+\ldots 1$ ($n$ times). If $n 1$ is never 0, we say that $R$ has characteristic 0.

A subring of a ring $R$ is a subset $S$ of $R$ that forms a ring under the operations of addition and multiplication defined on $R$. Subring of a ring $R$ are subsets of $R$ that inherit a ring structure from $R$.

子环¹

定义: A subring of a ring $R$ [with identity] is a subset $S$ of $R$ such that $S$ is a subgroup of $(R,+)$, is closed under multiplication $(x, y \in S$ implies $x y \in S)$, and ==contains the identity element==.

同态

定义: If $f: R \rightarrow S$, where $R$ and $S$ are rings, we say that $f$ is a ring homomorphism if $f(a+b)$ $=f(a)+f(b)$ and $f(a b)=f(a) f(b)$ for all $a, b \in R$, and $f\left(1_R\right)=1_S$.

理想

定义: Let $I$ be a subset of the ring $R$, and consider the following three properties:

$I$ is an additive subgroup of $R$.
If $a \in I$ and $r \in R$ then $r a \in I$; in other words, $r I \subseteq I$ for every $r \in R$.
If $a \in I$ and $r \in R$ then $a r \in I$; in other words, $I r \subseteq I$ for every $r \in R$.

If (1) and (2) hold, $I$ is said to be a left ideal of $R$.

If (1) and (3) hold, $I$ is said to be a right ideal of $R$.

If all three properties are satisfied, $I$ is said to be an ideal (or two-sided ideal) of $R$, a proper ideal if $I=R$, a nontrivial ideal if $I$ is neither $R$ nor $\{0\}$.

If $f: R \rightarrow S$ is a ring homomorphism, its kernel is

$\operatorname{ker} f=\{r \in R: f(r)=0\}$

$f$ is injective if and only if $\operatorname{ker} f=\{0\}$.

定义: Let $I$ be a proper ideal of the ring $R$. Since $I$ is a subgroup of the additive group of $R$, we can form the quotient group $R / I$, consisting of cosets $r+I$, $r \in R$. We define multiplication of cosets in the natural way:

$(r+I)(s+I)=r s+I.$

The cosets of $I$ form a ring, called the quotient ring of $R$ by $I$.

The identity element of the quotient ring is $1_R+I$, and the zero elemtent is $0_R+I$.

定义:
If $X$ is a non empty subset of the ring $R$, then $(X)$ will denote the ideal generated by $X$ , that is, the smallest ideal of $R$ that contains $X$. Explicitly,

$\langle X\rangle=R X R= \text{ the collection of all finite sums of the form } \sum_i r_i x_i s_i.$

with $r_i, s_i \in R$ and $x_i \in X$.

If $R$ is commutative, then $r x s=r s x$, and we may as well drop the $s$. In other words: In a commutative ring,

$\langle X\rangle=R X= \text{ all finite sums } \sum_i r_i x_i, r i \in R, x_i \in X.$

An ideal generated by a single element $a$ is called a principal ideal and is denoted by $\langle a\rangle$ or $(a)$.

定义: In an arbitrary ring, we will sometimes need to consider the sum of two ideals $I$ and $J$, defined as $\{x+y: x \in I, y \in J\}$. It follows from the distributive laws that $I+J$ is also an ideal. Similarly, the sum of two left [resp. right] ideals is a left [resp. right] ideal.

同余, 互素, 最大公因子, 最小公倍数

定义: If $a$ and $b$ are integers that are congruent modulo $n$, then $a-b$ is a multiple of $n$. Thus $a-b$ belongs to the ideal $I_n$ consisting of all multiples of $n$ in the ring $\mathbb{Z}$ of integers. Thus we may say that $a$ is congruent to $b$ modulo $I_n$. In general, if $a, b \in R$ and $I$ is an ideal of $R$, we say that $a \equiv b \bmod I$ if $a-b \in I$.

The integers $a$ and $b$ are relatively prime if and only if the integer $1$ can be expressed as a linear combination of $a$ and $b$. Equivalently, the sum of the ideals $I_a$ and $I_b$ is the entire ring $\mathbb{Z}$. In general, we say that the ideals $I$ and $J$ in the ring $R$ are relatively prime if $I+J=R$.

定义: Let $A$ be a nonempty subset of $R$, with $0 \notin A$. The element $d$ is a greatest common divisor (gcd) of $A$ if $d$ divides each $a$ in $A$, and whenever $e$ divides each $a$ in $A$, we have $e \mid d$.

The elements of $A$ are said to be relatively prime (or the set $A$ is said to be relatively prime) if $1$ is a greatest common divisor of $A$.

The nonzero element $m$ is a least common multiple ( $\mathrm{lcm})$ of $A$ if each $a$ in $A$ divides $m$, and whenever $a \mid e$ for each $a$ in $A$, we have $m \mid e$.

直积

定义: If $R_1, \ldots, R_n$ are rings, the direct product of the $R_i$ is defined as the ring of $n$ tuples $\left(a_1, \ldots, a_n\right)$, $a_i \in R_i$, with componentwise addition and multiplication, that is, with

$\begin{align*} \left(a_1, \ldots, a_n\right)+\left(b_1, \ldots, b_n\right)&=\left(a_1+b_1, \ldots, a_n+b_n\right) \\ \left(a_1, \ldots, a_n\right)\left(b_1, \ldots, b_n\right)&=\left(a_1 b_1, \ldots, a_n b_n\right). \end{align*}$

极大理想

定义: A maximal ideal in the ring $R$ is a proper ideal that is not contained in any strictly larger proper ideal.

单位, 不可约元, 素元

定义: A unit in a ring $R$ is an element with a multiplicative inverse. The elements $a$ and $b$ are associates if $a=u b$ for some unit $u$.

Let $a$ be a nonzero nonunit; $a$ is said to be irreducible if it cannot be represented as a product of nonunits. In other words, if $a=b c$, then either $b$ or $c$ must be a unit.

Again let $a$ be a nonzero nonunit; $a$ is said to be prime if whenever $a$ divides a product of terms, it must divide one of the factors. In other words, if $a$ divides $b c$, then $a$ divides $b$ or $a$ divides $c$ ($a$ divides $b$ means that $b=a r$ for some $r \in R$).

唯一因子分解整环

定义: A unique factorization domain (UFD) is an integral domain $R$ satisfying the following properties:

(UF1): Every nonzero element $a$ in $R$ can be expressed as $a=u p_1 \ldots p_n$, where $u$ is a unit and the $p_i$ are irreducible.

(UF2): If $a$ has another factorization, say $a=v q_1 \ldots q_m$, where $v$ is a unit and the $q_i$ are irreducible, then $n=m$ and, after reordering if necessary, $p_i$ and $q_i$ are associates for each $i$.

主理想整环, 欧式整环

定义: A principal ideal domain (PID) is an integral domain in which every ideal is principal, that is, generated by a single element.

Let $R$ be an integral domain. $R$ is said to be a Euclidean domain (ED) if there is a function $\Psi$ from $R \setminus\{0\}$ to the nonnegative integers satisfying the following property:

If $a$ and $b$ are elements of $R$, with $b \neq 0$, then $a$ can be expressed as $b q+r$ for some $q, r \in R$, where either $r=0$ or $\Psi(r)<\Psi(b)$.

We can replace “$r=0$ or $\Psi(r)<\Psi(b)$” by simply “$\Psi(r)<\Psi(b)$” if we define $\Psi(0)$ to be $-\infty$.

In any Euclidean domain, we may use the Euclidean algorithm to find the greatest common divisor of two elements.
A Euclidean domain is automatically a principal ideal domain, as we now prove.

multiplicative set

定义: Let $S$ be a subset of the ring $R$; we say that $S$ is multiplicative if

(a) $0 \notin S$,
(b) $1 \in S$, and
(c) whenever $a$ and $b$ belong to $S$, we have $a b \in S$.

We can merge (b) and (c) by stating that $S$ is closed under multiplication, if we regard $1$ as the empty product.

If $S$ is a multiplicative subset of the commutative ring $R$, we define the following equivalence relation on $R \times S$:

$(a, b) \sim(c, d)$ if and only if for some $s \in S$ we have $s(a d-b c)=0$.

If $a \in R$ and $b \in S$, we define the fraction $\frac{a}{b}$ to be the equivalence class of the pair $(a, b)$. The set of all equivalence classes is denoted by $S^{-1} R$, and in view of what we are about to prove, is called the ring of fractions of $R$ by $S$. The term localization of $R$ by $S$ is also used.

不可约多项式

定义: If $R$ is an integral domain, we will refer to an irreducible element of $R[X]$ as an irreducible polynomial. Nowin $F[X]$, where $F$ is a field, the units are simply the nonzero elements of $F$. Thus in this case, an irreducible element is a polynomial of degree at least 1 that cannot be factored into two polynomials of lower degree.

A polynomial that is not irreducible is said to be reducible or factorable.

交换环¹

素理想²

定义: A prime ideal in a commutative ring $R$ is a proper ideal $P$ such that for any two elements

$a, b$ in $R$, $a b \in P$ implies that $a \in P$ or $b \in P$.

多项式环²

定义: In this section, all rings are assumed commutative. To see a good reason for this restriction, consider the evaluation map (also called the substitution map) $E_x$, where $x$ is a fixed element of the ring $R$. This map assigns to the polynomial $a_0+a_1 X+\ldots+a_n X_n$ in $R[X]$ the value $a_0+a_1 x+\ldots+a_n x^n$ in $R$. It is tempting to say that “obviously”, $E_x$ is a ring homomorphism, but we must be careful. For example,

$\begin{align*} E_x[(a+b X)(c+d X)] & =E_x\left(a c+(a d+b c) X+b d X^2\right)=a c+(a d+b c) x+ b d x^2, \\ E_x(a+b X) E_x(c+d X) & =(a+b x)(c+d x)=a c+a d x+b x c+b x d x, \end{align*}$

and these need not be equal if $R$ is not commutative.

The degree, abbreviated deg, of a polynomial $a_0+a_1 X+\ldots+a_n X^n$ (with leading coefficient $a_n \neq 0$) is $n$; it is convenient to define the degree of the zero polynomial as $-\infty$.

Primary Decomposition

定义: In a commutative ring $R$, the radical $\mathrm{Rad}\mathfrak{a}$ of an ideal $\mathfrak{a}$ is the intersection of all prime ideals of $R$ that contain $\mathfrak{a}$. If $\mathfrak{a}=R$, then no prime ideal of $R$ contains $\mathfrak{a}$ and we let the empty intersection $\mathrm{Rad}\mathfrak{a}$ be $R$ itself. In general, $\mathrm{Rad}\mathfrak{a}$ is sometimes denoted by $\sqrt{\mathfrak{a}}$.

定义: An ideal $\mathfrak{a}$ of a commutative ring $R$ is semiprime when it is an intersection of prime ideals.

定义: An ideal $\mathfrak{q}$ of a commutative ring $R$ is primary when $\mathfrak{q}\neq R$ and, for all $x,y\in R$, $xy\in\mathfrak{q}$ implies $x\in\mathfrak{q}$ or $y^{n}\in\mathfrak{q}$ for some $n>0$. An ideal $\mathfrak{q}$ of $R$ is $\mathfrak{p}$-primary when $\mathfrak{q}$ is primary and $\mathrm{Rad}\mathfrak{q}=\mathfrak{p}$.

Ring Extensions

定义: A ring extension of a commutative ring $R$ is a commutative ring $E$ of which $R$ is a subring.

交换环 $R$ 的环扩张ring extension $E$ 是一个交换环, 且使 $R$ 是它的子环.

定义: A ring extension $E$ of $R$ is finitely generated over $R$ when $E=$ $R\left[\alpha_{1},\ldots,\alpha_{n}\right]$ for some $n\geqq0$ and $\alpha_{1},\ldots,\alpha_{n}\in E$.

设 $E$ 为 $R$ 上的环扩张, 如果对于某个 $n\ge 0$, 和 $\alpha_1,\cdots,\alpha_n\in E$ 有 $E=R[\alpha_1,\cdots,\alpha_n]$, 则称 $E$ 为由 $R$ 有限生成的.

定义: Let $R$ be a ring. An $R$-module is an abelian group $M$ together with an action $(r,x)\longmapsto rx$ of $R$ on $M$ such that $r(x+y)=rx+ry$, $(r+s)x=rx+sx$, $r(sx)=(rs)x$, and $1x=x$, for all $r,s\in R$ and $x,y\in M$. A submodule of an $R$-module $M$ is an additive subgroup $N$ of $M$ such that $x\in N$ implies $rx\in N$ for every $r\in R$.

设 $R$ 是一个环, $M$ 是一个 Abel 群, 若 $R$ 在 $M$ 上的作用 $R\times M\to M$, $(r,x)\longmapsto rx$ 满足, 对于任意的 $r,s\in R$ 和 $x,y\in M$, 有 $r(x+y)=rx+ry$, $(r+s)x=rx+sx$, $r(sx)=(rs)x$, 和 $1x=x$, 称 $M$ 为 $R$-模.

$R$-模 $M$ 的子模是 $M$ 的加法子群 $N$, $N$ 满足对于任意的 $r\in R$, $x\in N$, 有 $rx\in N$.

定义: Let $M$ be an R-module. The submodule of $M$ generated by a subset $X$ of $M$ is the set of all linear combinations of elements of $X$ with coefficients in $R$. A submodule of $M$ is finitely generated when it is generated (as a submodule) by a finite subset of $M$.

定义: An element $\alpha$ of a ring extension $E$ of $R$ is integral over $R$ when it satisfies one of the following equivalent conditions:

(1) $f(\alpha)=0$ for some monic polynomial $f\in R[X]$;

(2) $R[\alpha]$ is a finitely generated submodule of $E$;

(3) $\alpha$ belongs to a subring of $E$ that is a finitely generated $R$-module.

Integral Extensions

定义: A ring extension $R\subseteq E$ is integral, and $E$ is integral over $R$, when every element of $E$ is integral over $R$.

如果 $E$ 中的每个元素在 $R$ 上是整的, 则称环扩张 $R\subseteq E$ 是整的, 且 $E$ 在 $R$ 上是整的,

定义: In a ring extension $E$ of $R$, an ideal $\mathfrak{A}$ of $E$ lies over an ideal $\mathfrak{a}$ of $R$ when $\mathfrak{A}\cap R=\mathfrak{a}$.

定义: The integral closure of a ring $R$ in a ring extension $E$ of $R$ is the subring $\overline{R}$ of $E$ of all elements of $E$ that are integral over $R$. The elements of $\overline{R}\subseteq E$ are the algebraic integers of $E$ (over $R$ ).

环 $R$ 在环扩张 $E$ 中的整闭包代数闭包的推广是 $E$ 中的子环 $\overline{R}$, 由 $E$ 中所有在 $R$ 上整的元素组成的集合. 即 $\overline{R}\subseteq E$ 中的元素是 $E$ 上 ($R$ 上) 的全体代数整数.

定义: A domain $R$ is integrally closed when its integral closure in its quotient field $Q(R)$ is $R$ itself (when no $\alpha\in Q(R)\backslash R$ is integral over $R$ ).

设 $S$ 是环 $R$ 的扩张, 若 $R$ 的整闭包就是 $R$ 本身, 即 $\overline{R}=R$, 则称 $R$ 在 $S$ 上整闭integrally closed.

Localization

定义: A multiplicative subset of a commutative ring $R$ is a subset $S$ of $R$ that contains the identity element of $R$ and is closed under multiplication. A multiplicative subset $S$ is proper when $0\notin S$.

定义: If $S$ is a proper multiplicative subset of a commutative ring $R$, then $S^{-1}R$ is the ring of fractions of $R$ with denominators in $S$.

定义: Let $S$ be a proper multiplicative subset of $R$. The contraction of an ideal $\mathfrak{A}$ of $S^{-1} R$ is $\mathfrak{A}^C=\{a \in R \mid a / 1 \in \mathfrak{A}\}$. The expansion of an ideal $\mathfrak{a}$ of $R$ is $\mathfrak{a}^E=\left\{a / s \in S^{-1} R \mid a \in \mathfrak{a}, s \in S\right\}$.

定义: The localization of a commutative ring $R$ at a prime ideal $\mathfrak{p}$ is the ring of fractions $R_{\mathfrak{p}}=(R \backslash \mathfrak{p})^{-1} R$.

定义: A commutative ring is local when it has only one maximal ideal.

局部环 $R$ (或$(R,\mathfrak{m})$) 是只有一个极大理想 $\mathfrak{m}$ 的(含幺)交换环. 域 $R/\mathfrak{m}$ 称为 $R$ 的剩余域.
若 $R$ 中仅有有限个极大理想, 则称为半局部环.
一个局部环 $(R,\mathfrak{m})$ 上带有一个自然的 $\mathfrak{m}$-进拓扑, 使得 $R$ 称为拓扑环; 拓扑中的开集由 $\{\mathfrak{m}^i:i\ge0\}$ 生成. 当 $R$ 为诺特环时, 可证明 $R$ 是 Hausdorff 空间, 且所有理想都是闭理想.
设 $(R,\mathfrak{m}), (S,\mathfrak{n})$ 为局部环, 环同态 $\phi:R\to S$ 称为是局部同态, 当且仅当 $\mathfrak{m}=\phi^{-1}(\mathfrak{n})$.

Dedekind Domains

定义: A fractional ideal of a domain $R$ is a subset of its quotient field $Q$ of the form $\mathfrak{a} / c=\{a / c \in Q \mid a \in \mathfrak{a}\}$, where $\mathfrak{a}$ is an ideal of $R$ and $c \in R$, $c \neq 0$.

分式理想分式理想是比理想更广的一种理想. 它由整环上的分式域或商域构成.是整环 $R$ 的分式域 $Q$ 的子集, 是形如 $\mathfrak{a} / c=\{a / c \in Q \mid a \in \mathfrak{a}\}$ 的集合, 其中 $\mathfrak{a}$ 是 $R$ 的理想, $c \in R$, $c \neq 0$.

定义: A fractional ideal $\mathfrak{A}$ of $R$ is invertible when $\mathfrak{A} \mathfrak{B}=R$ for some fractional ideal $\mathfrak{B}$ of $R$.

在整环 $R$ 中的分式理想 $\mathfrak{A}$, 若存在某个分式理想 $\mathfrak{B}$, 使得 $\mathfrak{A} \mathfrak{B}=R$, 则称 $\mathfrak{A}$ 是可逆的.

定义: Dedekind domains are defined by the following equivalent conditions.

every nonzero ideal of $R$ is invertible (as a fractional ideal);
every nonzero fractional ideal of $R$ is invertible;
every nonzero ideal of $R$ is a product of prime ideals of $R$;
every nonzero ideal of $R$ can be written uniquely as a product of positive powers of distinct prime ideals of $R$.

Dedekind 整环 Dedekind整环是一维诺特整闭环是满足以下四个等价条件之一的整环:

$R$ 的任何非零理想是可逆的;

$R$ 的任何非零分式理想是可逆的;

$R$ 的任何非零理想是 $R$ 中素理想的乘积;

$R$ 的任何非零理想可以唯一的表示为 $R$ 中不同素理想的整数幂之积.

整环 $R$ 中的 Dedekind 整环是满足以下三个条件的整环:

Noetherian,

整闭,

所有非零素理想是极大理想.

Krull Dimension

定义: In a commutative ring $R$, the height $\mathrm{hgt} \mathfrak{p}$ of a prime ideal $\mathfrak{p}$ is the least upper bound of the lengths of strictly descending sequences $\mathfrak{p}=\mathfrak{p}_0 \supsetneqq$ $\mathfrak{p}_1 \supsetneqq \cdots \supsetneqq \mathfrak{p}_m$ of prime ideals of $R$.

定义: The spectrum of a commutative ring is the set of its prime ideals, partially ordered by inclusion. The Krull dimension or dimension $\mathrm{dim} R$ of $R$ is the least upper bound of the heights of the prime ideals of $R$.

Algebraic Sets

定义: Let $K$ be a field and let $\overline{K}$ be its algebraic closure. The zero set of a set $S \subseteq K\left[X_1, \ldots, X_n\right]$ of polynomials is

$Z(S)=\left\{x=\left(x_1, \ldots, x_n\right) \in \overline{K}^n \mid f(x)=0 \text { for all } f \in S\right\} .$

An algebraic set in $\overline{K}^n$ with coefficients in $K$ is the zero set of a set of polynomials $S \subseteq K\left[X_1, \ldots, X_n\right]$.

定义: An algebraic set $A \subseteq \overline{K}^n$ is irreducible, or is an algebraic variety, when $A \neq \varnothing$ and $A$ is the zero set of a prime ideal.

定义: The dimension $\mathrm{dim} A$ of an algebraic variety $A \subseteq \overline{K}^n$ is the length of the longest strictly decreasing sequence $A=A_0 \supsetneqq A_1 \supsetneqq \cdots \supsetneqq A_r$ of nonempty algebraic varieties contained in A; equivalently, $n-\mathrm{hgt} \mathfrak{I}(A)$.

定义: The Zariski topology on an algebraic set $A \subseteq \overline{K}^n$ is the topology whose closed sets are the algebraic sets $B \subseteq A$.

Regular Mappings

定义: Let $A \subseteq \overline{K}^n$ be an algebraic set. A polynomial function of $A$ is a mapping of $A$ into $\overline{K}$ that is induced by a polynomial $f \in K\left[X_1, \ldots, X_n\right]$. The coordinate ring of $A$ is the ring $C(A)$ of all such mappings.

定义: Let $K$ be a field and let $A \subseteq \overline{K}^m, B \subseteq \overline{K}^n$ be algebraic varieties. A mapping $F=\left(f_1, \ldots, f_n\right): A \longrightarrow B \subseteq \overline{K}^n$ is regular when its components $f_1, \ldots, f_n: A \longrightarrow \overline{K}$ are polynomial functions.

定义: Two algebraic varieties $A$ and $B$ are isomorphic when there exist mutually inverse regular bijections $A \longrightarrow B$ and $B \longrightarrow A$.