minimal left ideals

**dori1123** · June 4th 2010, 10:48 AM

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

I know how to prove that $Col(l)$ is a left ideal in $Mat_n(K)$ , but having trouble with the minimality part of the proof. Can I get some help please?

**NonCommAlg** · June 4th 2010, 04:14 PM

Originally Posted by dori1123

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

I know how to prove that $Col(l)$ is a left ideal in $Mat_n(K)$ , but having trouble with the minimality part of the proof. Can I get some help please?

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

$1 \leq r \leq n,$ and hence $I=Col(l)$ because $\{e_{rl}: \ 1 \leq r \leq n \}$ is a $K$ basis for $Col(l).$

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