If the health of the mother or the health of the child is threatened, the parent is exempt from fasting. If the mother is so exempted from fasting during the days of Ramadan, those days must be made up for at a latter time when she is able.
If a breastfeeding mother and her child are both healthy such that the mother is able to fast in Ramadan without harming herself of the child, then fasting for her is a duty.