On the equation -= in division rings

PROOF. If R satisfies (C) then, for each nonzero element a in R the equations ax = c and xa = c have solutions x in R, and so R is a division ring as is shown in many places. (For example, see page 87 of [7].) In view of this lemma our question reduces to QUESTION 1. Can a noncommutative division ring satisfy condition (C)? The following lemma shows that if condition (C) is slightly modified, then the answer to the corresponding question is "no."