New York City’s Correction Department, facing an outcry over the death of a mentally ill inmate in an overheated cell at Rikers Island, announced on Thursday that it was demoting the warden in charge of the mental health unit and punishing two other employees.

The inmate, Jerome Murdough, died in February after being left unattended for hours as the temperature in his cell climbed above 100 degrees. The authorities blamed a faulty heating unit for the high temperatures and said a correction officer who should have been checking regularly on inmates failed to do so.

Mr. Murdough’s death has intensified concerns about the city’s ability to care for prisoners who are mentally ill, particularly at Rikers Island, where inmates with mental illness are frequently neglected and abused. Mr. Murdough, 56, a homeless veteran with a history of alcohol abuse, died only a week after arriving at Rikers. He had been arrested on Feb. 7 on a trespassing charge.

In demoting the warden, the Correction Department acknowledged “systemic management problems” at the Anna M. Kross Center, the mental health unit where Mr. Murdough died, a statement said. The warden there, Rose Agro, will be put in charge of a jail that does not house those who are mentally ill, the department said.