We define a ring R to be right cP- Baer if the right annihilator of a cyclic projective right R-module is generated by an idempotent. We also define a ring R to be right I-extending if each ideal generated by an idempotent is right essential in a direct summand of R. It is shown that the two conditions are equivalent in a semiprime ring. Next we define a right I-prime ring, which generalizes the prime condition. This condition is equivalent to all cyclic projective right R-modules being faithful. For a semiprime ring, we show the existence of a cP -Baer hull. We also provide some results about the p.q.-Baer hull and when it is equal to the cP-Baer hull. Polynomial and formal power series rings are studied with respect to the right cP-Baer condition. In general, a formal power series ring over one indeterminate in which its base ring is right p.q.-Baer ring is not necessarily right p.q.-Baer. However, if the base ring is right cP-Baer then the formal power series ring over one indeterminate is right cP-Baer. The fifth chapter is devoted to matrix extensions of right cP-Baer rings. A characterization of when a 2-by-2 generalized upper triangular matrix ring is right cP-Baer is given. The last major theorem is a decomposition of a cP-Baer ring, satisfying a finiteness condition, into a generalized triangular matrix ring with right I-prime rings down the main diagonal. Examples illustrating and delimiting our results are provided.