Do you mean here that I is an ideal of R? That certainlyisa ring. Whether it is a division ring depends upon whether all elements have a multiplicative inverse. Of course, if R itself is a division ring, R/I is. If R is not a division ring then it will have "zero divisors"- that is, there will exist a and b, neither 0, such that ab= 0. In that case (a+ i)(b+ I)= ab+I= I. That would, I think, imply that R/I is a division ring if and only if all zero-divisors are in I.