→ recorriendo :感謝 09/27 23:45
※ 引述《recorriendo (孟新)》之銘言：
: If R is a finite dimensional algebra over a field k and if rs = 1, show
: that sr = 1. Deduce that if R contains no zero-divisors, then every non-zero
: element of R has a two sided inverse, that is, R is a division algebra over k.
: 這要怎麼證呢
: 好像沒有很難卻一直想不到
: 還有前後半部的關係也看不出來
: 求教高手
: 感謝
前面有人弄了，我只弄後面：
Consider the k-linear map T:R → R, by T(x)=r(x).
"R has no zero divisors" implies "T is an injective linear map"
Since R is finite dimensional over k, T is then surjective.
There exists an s such that T(s)=rs=1.
The first part says that s is also a left inverse.
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