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, Ker f ⊆ I P, and whenever P = Ker f + 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 Z (_RR), Soc (_RR), δ (_RR) and Z₂ (_RR).