We say a ring R is pi-Baer if the right annihilator of every projection invariant left ideal of R is generated by an idempotent element of R. In this paper, we study connections between the pi-Baer condition and related conditions such as the Baer, quasi-Baer and pi-extending conditions. The 2-by-2 generalized triangular and the n-by-n triangular pi-Baer matrix rings are characterized. Also, we prove that a n-by-n full matrix ring over a pi-Baer ring is a pi-Baer ring. In contrast to the Baer condition, it is shown that the pi-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.