抽象代数入门（一）

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本文探讨了数学中抽象代数的基本概念，包括群论、环论等，详细解析了幺半群、群、环的定义及其特性，如二元运算、单位元、逆元、加法与乘法的结合性和交换性等。

一、群论

幺半群(monoid)

之前看老师讲的叫Abelian monoid（阿贝尔幺半群），但是搜不到。

A monoid is a set closed under an associative binary operation and has an identity element $\mathbf{I}\in \mathbf{S}$ such that $\forall \mathbf{a}\in\mathbf{S}, \mathbf{a}\mathbf{I}=\mathbf{I}\mathbf{a}=\mathbf{a}$ .

Binary operation(二元运算): f(x, y) an operation between two quantities or expressions x and y.

An binary operation on a nonempty set $\mathbf{A}$ is a map of $f: \mathbf{A}\times\mathbf{A}\rightarrow\mathbf{A}$ .such that:

$f(v_1, v_2)=f(v_2, v_1)$ , "abelian commutative";
$f(f(v_1, v_2), v_3) = f(v_1, f(v_2, v_3))$ , "abelian associative";
$\exists e\in\mathbf{A},s.t. f(v, e)=v, \forall v\in \mathbf{A}$ , e is the "identity element".同上面的 $\mathbf{I}$ .

常用的二元运算有：addition $+$ , substraction $-$ , multiplication $\times$ , division $\div$ .

不同于群（group），幺半群中的元素不一定有逆。

群(group)

也叫Abelian group.在 monoid 的基础上多了一个条件:

Every element has an inverse: $\forall g\in \mathbf{A}, \exists h\in \mathbf{A}, s.t. f(g, h)= f(h, g)= e$ , where e is the identity element.

例子：

Every vector space $(\mathbf{V}, *)$ is Abelian group.

$v\in \mathbf{V}, -v\in\mathbf{V}, v + (-v)=0\in\mathbf{V}$ . 这里 0 是单位元素(identity element).

二、环论（ring)

A ring in the mathmatical sense is a set $\mathbf{S}$ together with two binary operations $+, *$ (分表加法和乘法). 满足以下条件：

加法结合性(additive assocativity): for all a, b, c in S, s.t. (a+b)+c = a + (b+c);
加法交换性(additive commutativity): for all a, b in S, s.t. a + b = b + a;
加法单位元(additive identity): there exists an element e in S, s.t. a + e = e + a = a, 这个 e 也可以写成 0, 加法的单位元是0;
分配性(left and right distributivity): for all a, b, c in S, s.t. a * (b+c) = a*b + a*c,以及 (a + b) *c = a*c + b*c;
加法可逆(additive inverse): for every a in S, there exists -a in S, s.t. a + (-a) = e, e 同上， e = 0.
乘法结合性(multiplicative associativity)： $\forall a,b,c\in \mathbf{S}, s.t. a*(b*c)=(a*b)*c$ 。满足该性质的环也叫 associative ring.
乘法交换性(mulitplicative commutativity): $\for a,b \in \mathbf{S}, s.t. a*b = b*a$ . (a ring satisfying this property is termed a commutative ring).
乘法单位元(multiplicative identity): There exists an element $1\in\mathbf{S}, s.t. 1*a=a*1,\forall a\in \mathbf{S}$ .(a ring satisfying this property is termed a unit ring, or a ring with identity).
乘法可逆(multiplicative inverse): $\forall a\in\mathbf{S}, a \not= 0, \exists a^{-1}\in\mathbf{S}, s.t. a * a^{-1}=a^{-1}*a=1$ . Where 1 is the identity element.