Re: [代數] ringR的left ideal只有(0)跟R

[LimSinE (r=e^theta)]

※ 引述《ma4wanderer (台師怪客)》之銘言：
: 這題是說ring R 的left ideal 只有R跟(0)
: 則
: 1.R 是division ring
: 或
: 2.has p elements(p is a prime) and ab=0    for all a、b in R
: 有個人跟我講考慮這個ideal {r in R| rR=0}
: 可是除環的1不知怎麼弄出來
: 還有prime elements哪裡來
: ＠＠
: 有請指點!

1. Ra is a left ideal = R or 0

2. {a| Ra=0} is a left ideal = R or 0
If R, then xy=0 for all x,y. left ideal = (normal) additive subgroup.
=> cyclic group of order p.

If 0, then Ra=R for all a=/=0

3.For a=/=0
{r|ra=0} is a left ideal = R or 0
If R, then Ra = 0, impossible
So ra=0 => r=0
conclusion: xy=0 => x=0 or y=0, left and right cancellations.

4.
For a=/=0, Ra=R, so there exists e_a,  e_a a=a
For any b, b e_a a = ba, so b e_a = b by cancellation.

Now pick a, b, c=/=0, then b e_a = b = be_c, so e_a = e_c by cancellation.
Denote the common element e_a by e, the left and right identity.

5.
For a=/=0, Ra=R, so there exists a', a'a=e
Then a'aa' =ea' = a'e so aa'=e by cancellation.
And a' is the multiplicative inverse of a.



--
r=e^theta
即使有改變，我始終如一。

--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 219.84.3.42
推 yusd24      :讓我膜拜一下
推 chy1010     :    膜拜一下
推 ma4wanderer :"cyclic" group of order "p" 這邊看無QQ
推 bineapple   :Sylow's first theorem
推 ma4wanderer :我好像catch到了
推 ma4wanderer :太神奇了QQ