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