# Thread: question on matrix ring over a field

1. ## question on matrix ring over a field

Let
R = Mn(F) the ring consists of all n*n matrices over a finite 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). Then the following hold:
1.
RE is a maximal left ideal.
2. If
A is a rank n-1 matrix in R then A is similar to E.

what is the proof of the above statements?
thank you

2. Originally Posted by xixi

Let R = Mn(F) the ring consists of all n*n matrices over a finite 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). 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.

3. ## Thank you

Thank you very much ,you're really the Lord of the Rings ,what do you think of the second statement , mustn't it be "If A is a rank n-1 matrix in RE then A is similar to E "?

4. Originally Posted by xixi

what do you think of the second statement , mustn't it be "If A is a rank n-1 matrix in RE then A is similar to E "?
if $\displaystyle A$ has rank $\displaystyle n-1,$ then one column (we may assume it's the last column) is a linear combination of other columns. then doing some column operations, we can make the entries of the last

column all zero. so every matrix of rank $\displaystyle n-1$ is similar to an element of $\displaystyle RE.$ (note that $\displaystyle RE$ is exactly the set of all elements of $\displaystyle R$ whose last columns are zero). so yes, you may assume that

$\displaystyle A \in RE.$

5. ## how do you prove similarity?

we should prove that A is similar to E , i.e there must be an invertible matrix P such that PA(P^-1) = E , how would you say this?