When is a hospital liable for its physicians?
As a general rule of law, the private, charitable hospital bears no liability to the patient who is injured in the hospital by a physician while being treated as that physician's private patient.  However, there are exceptions to this rule. Ever since the Darling decision, the courts have been imposing more and more responsibility upon hospital administration, and ultimately the board of trustees, for acts occurring within the hospital's walls.
Morris
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D006075:Governing Board; D006739:Hospital Administration / Q000331:legislation & jurisprudence; D006788:Hospitals, Voluntary / Q000331:legislation & jurisprudence*; D007882:Legislation, Hospital; D008318:Malpractice; D008505:Medical Staff, Hospital / Q000331:legislation & jurisprudence*