# Thread: Degree of a matrix

1. ## Degree of a matrix

Let $R=M_n(F)$ where $F$ is a field . Suppose that $E_{ij}$ is a matrix whose $(i,j)$th entry is 1 and the others are 0 . For any element $A\in R$ , the number of elements of $R$ such as $B$ for which we have $RA+RB=R$ is the degree of $A$ . Now prove that $E_{11}$ has the minimum degree in $R-${ $0$} .

2. Originally Posted by xixi
Let $R=M_n(F)$ where $F$ is a field . Suppose that $E_{ij}$ is a matrix whose $(i,j)$th entry is 1 and the others are 0 . For any element $A\in R$ , the number of elements of $R$ such as $B$ for which we have $RA+RB=R$ is the degree of $A$ . Now prove that $E_{11}$ has the minimum degree in $R-${ $0$} .
i wanted to ignore this question because, as usual, there's something wrong with your question: if F is an infinite field and B is any unit of R, then RB = R and so RA + RB = R.

that means the degree, as you defined, of every element of R is infinity (!!) because R will have infinitely many units.

so my guess is that the $+$ might be $\oplus$ or the degree might be the number of "left ideals" RB, and not the number of "matrices B", such that RA + RB = R.

by the way, are these problems from a textbook? do you usually change them?

3. Originally Posted by NonCommAlg
i wanted to ignore this question because, as usual, there's something wrong with your question: if F is an infinite field and B is any unit of R, then RB = R and so RA + RB = R.

that means the degree, as you defined, of every element of R is infinity (!!) because R will have infinitely many units.

so my guess is that the $+$ might be $\oplus$ or the degree might be the number of "left ideals" RB, and not the number of "matrices B", such that RA + RB = R.

by the way, are these problems from a textbook? do you usually change them?
Yeah , sorry , it wasn't my fault , I thought that I have included all the assumptions but I dropped that $F$ is a finite field .