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**xixi** Let *R *= *Mn*(*F*) the ring consists of all *n*n *matrices over a finite field *F *and *E *= *E*11 + *E*22 + ... + *En-*1,*n-*1,

where *Eii *is the elementary matrix(*Eij is matrix whose ij th element is 1 and the others are 0*). Then the following hold:

1. *RE *is a maximal left ideal.

the result holds for infinite fields as well as finite fields. let $\displaystyle I$ be a left ideal of $\displaystyle R$ which properly contains $\displaystyle RE$ and choose $\displaystyle [a_{ij}]=A \in I \setminus RE.$ since $\displaystyle A \notin RE,$ there must exist $\displaystyle 1 \leq k \leq n$

such that $\displaystyle a_{kn} \neq 0.$ see that $\displaystyle E_{nn}=a_{kn}^{-1}E_{nk}(A-AE) \in I$ and thus $\displaystyle 1_R=E+E_{nn} \in I.$ therefore $\displaystyle I=R$ and the proof is complete.