Generalized /P-Injective Rings

For a ring R, let ip(R_R) = {α ∈ R: every right R-homomorphism f from any right ideal of R into R with Imf = αR can extend to R}. It is known that R is right IP-injective if and only if R = ip(R_R) and R is right simple-injective if and only if {α ∈ R : αR is simple} (?) ip(R_R). In this note, we introduce the concept of right S-IP-injective rings, i.e., the ring R with S (?) ip(R_R), where S is a subset of R. Some properties of this kind of rings are obtained.