Is the money I gave my mother and relinquished to her, but which she refused to take ownership of and said she would keep for me, considered my property or that of the heirs after her death?

A gift is established by offer and acceptance, and becomes binding upon possession according to the majority of scholars. If acceptance does not occur, the gift is invalid. If one of the contracting parties dies before acceptance, the contract is nullified. If your mother did not accept the gift, but rather explicitly rejected it and stated that she was taking possession of it as a trust, then the gift was not completed. It remains your property and does not belong to the heirs.