# Thread: [SOLVED] maximal left ideal or not?

1. ## [SOLVED] maximal left ideal or not?

Let R = Mn(F) be the ring consists of all n*n matrices over a field F and
E = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix (Eij is matrix whose ij th element is 1 and the others are 0) .
I know that RE is a maximal left ideal . Let Q be an invertible matrix . Can we say that REQ is a maximal left ideal ?

2. Originally Posted by xixi
Let R = Mn(F) be the ring consists of all n*n matrices over a field F and
E = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix (Eij is matrix whose ij th element is 1 and the others are 0) .
I know that RE is a maximal left ideal . Let Q be an invertible matrix . Can we say that REQ is a maximal left ideal ?
yes. if $\displaystyle REQ \subseteq I \lhd_{\ell} R,$ then $\displaystyle RE \subseteq IQ^{-1}.$ now $\displaystyle IQ^{-1}$ is a left ideal of $\displaystyle R$ and so, since $\displaystyle RE$ is a maximal left ideal, we must have either $\displaystyle IQ^{-1}=R$ or $\displaystyle IQ^{-1}=RE.$

thus either $\displaystyle I=RQ=R$ or $\displaystyle I=REQ.$