Rich joined Emap, which was acquired by its German rival Bauer in December last year, more than 25 years ago and has held a variety of senior roles at the media company.
Bauer said that Rich's departure is a result of the company looking to simplify the structure and that he is leaving by mutual agreement.
Rich will not be replaced, but the Bauer Advertising and London Lifestyle businesses will now report directly to Bauer Consumer Media chief executive Paul Keenan.
Rich's previous roles with the company include managing director of Emap Advertising and managing director of FHM Worldwide.
He is the latest senior executive to depart Bauer, following Dharmash Mistry, the head of its digital operations, and Mark Frith, editor-in-chief of Heat magazine.
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