A generalization of projective covers

Let M be a left module over a ring R and I an ideal of R. We call (P,f) a projective I-cover of M if f is an epimorphism from P to M, P is projective, Kerf IP, and whenever P=Kerf+X, then there exists a summand Y of P in Kerf such that P=Y+X. This definition generalizes projective covers and projective δ-covers. Similar to semiregular and semiperfect rings, we characterize I-semiregular and I-semiperfect rings which are defined by Yousif and Zhou using projective I-covers. In particular, we consider certain ideals such as , , and .