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 $\displaystyle +$ might be $\displaystyle \oplus$ 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?