Re: [代數] ringR的left ideal只有(0)跟R
※ 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。
--
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