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

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?