We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator l R (Ra) n is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characte...