抽象代數入門（一）

一、羣論

幺半羣(monoid)

之前看老師講的叫Abelian monoid（阿貝爾幺半羣），但是搜不到。

A monoid is a set closed under an associative binary operation and has an identity element  such that .

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

An binary operation on a nonempty set  is a map of .such that:

1. , "abelian commutative";
2. , "abelian associative";
3. , e is the "identity element".同上面的 .

常用的二元運算有：addition , substraction , multiplication , division .

不同於羣（group），幺半羣中的元素不一定有逆。

 

 

羣(group)

也叫Abelian group.在 monoid 的基礎上多了一個條件:

Every element has an inverse: , where e is the identity element.

例子：

Every vector space  is Abelian group. 

. 這裏 0 是單位元素(identity element).

二、環論（ring)

A ring in the mathmatical sense is a set  together with two binary operations (分表加法和乘法). 滿足以下條件：

1. 加法結合性(additive assocativity): for all a, b, c in S, s.t. (a+b)+c = a + (b+c);
2. 加法交換性(additive commutativity): for all a, b in S, s.t. a + b = b + a;
3. 加法單位元(additive identity): there exists an element e in S, s.t. a + e = e + a = a, 這個 e 也可以寫成 0, 加法的單位元是0;
4. 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;
5. 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.
6. 乘法結合性(multiplicative associativity)：。滿足該性質的環也叫 associative ring.
7. 乘法交換性(mulitplicative commutativity): . (a ring satisfying this property is termed a commutative ring).
8. 乘法單位元(multiplicative identity): There exists an element .(a ring satisfying this property is termed a unit ring, or a ring with identity).
9. 乘法可逆(multiplicative inverse): . Where 1 is the identity element.

定義一個環至少要滿足前五個條件，但一般會加上第六個條件變成associative ring. 沒有滿足第六個條件的叫nonassociative ring.

滿足所有條件的就稱爲域(field).不滿足 7 乘法交換的環叫做 division algebra(可除代數)也叫 skew field(非交換域).