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Math Help - minimal left ideals

  1. #1
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    minimal left ideals

    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?
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    Quote Originally Posted by dori1123 View Post
    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).
    Last edited by NonCommAlg; June 4th 2010 at 05:24 PM.
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