Sharon has caused diarrhea in her three-year-old daughter by feeding her spoiled milk and rotten eggs. When the child is repeatedly hospitalized, Sharon acts like a very worried and caring mother. Because Sharon does not get to play the sick role and gets no obvious reward for her behavior, her appropriate diagnosis is .

Factitious disorder imposed on another

Explanation:

Factitious disorder imposed on another, which was usually referred to as Munchausen syndrome by proxy, is characterized by a deliberate initiation or falsely claiming of illness on someone, usually on a dependent  a caregiver, causing the dependent to be admitted to be given medical care on such claims of being ill.

As stated in the question, Sharon created the appearance of diarrhea in her child, to get her admitted while pretending to be a worried and caring mother. Factitious disorder imposed on another is the appropriate diagnosis for Sharon.