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Thread: minimal left ideals

  1. #1
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    minimal left ideals

    Let $\displaystyle K$ be a field. For any $\displaystyle l \in \{1,2,...,n\}$, let $\displaystyle Col(l)=\{A=[a_{ij}] \in Mat_n(K): a_{ij}=0, \forall j \not= l\}$. Prove that $\displaystyle Col(l)$ is a minimal left ideal in $\displaystyle Mat_n(K)$.

    I know how to prove that $\displaystyle Col(l)$ is a left ideal in $\displaystyle Mat_n(K)$, but having trouble with the minimality part of the proof. Can I get some help please?
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Let $\displaystyle K$ be a field. For any $\displaystyle l \in \{1,2,...,n\}$, let $\displaystyle Col(l)=\{A=[a_{ij}] \in Mat_n(K): a_{ij}=0, \forall j \not= l\}$. Prove that $\displaystyle Col(l)$ is a minimal left ideal in $\displaystyle Mat_n(K)$.

    I know how to prove that $\displaystyle Col(l)$ is a left ideal in $\displaystyle Mat_n(K)$, but having trouble with the minimality part of the proof. Can I get some help please?
    let $\displaystyle I \neq (0)$ be a left ideal of $\displaystyle M_n(K)$ which is contained in $\displaystyle Col(l),$ for some $\displaystyle 1 \leq l \leq n.$ i'll prove that $\displaystyle I=Col(l).$ let $\displaystyle 0 \neq x = \sum_{i=1}^n a_i e_{il} \in I,$ where $\displaystyle a_i \in K$ and $\displaystyle e_{ij}$ is an element of $\displaystyle M_n(K)$ which has 1 in the $\displaystyle (i,j)$ position and 0 elsewhere. so there exists some $\displaystyle 1 \leq k \leq n$ such that $\displaystyle a_k \neq 0.$ now let $\displaystyle 1 \leq r \leq n$. then, since $\displaystyle I$ is a left ideal of $\displaystyle M_n(K)$, we have $\displaystyle a_k^{-1}e_{rk}x = e_{rl} \in I.$ so $\displaystyle e_{rl} \in I,$ for all

    $\displaystyle 1 \leq r \leq n,$ and hence $\displaystyle I=Col(l)$ because $\displaystyle \{e_{rl}: \ 1 \leq r \leq n \}$ is a $\displaystyle K$ basis for $\displaystyle Col(l).$
    Last edited by NonCommAlg; Jun 4th 2010 at 04:24 PM.
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