
Ring?
I'm not sure if we can consider this a ring, but could it be a division ring?
Suppose + and * are well defined (* is multiplication).
Is < R/I, +, * > a ring, if R/I = {a + I: a is an element of R}?
I understand that + means (a + I) + (b + I) = (a + b) + I and * means (a + I) (b + I) = ab + I

Do you mean here that I is an ideal of R? That certainly is a 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 zerodivisors are in I.