حلقه هایی که مدولهای دوری آن به صورت اساسی در مدولهای تصویری نشانده می شوند

‏‎for the first time nakayama introduced qf-ring. in 1967 carl. faith and elbert a. walker showed that r is qf-ring if and only if each injective right r-module is projective if and only if each injective left r-modules is projective. in 1987 s.k.jain and s.r.lopez-permouth proved that every ring homomorphic images of r has the property that each cyclic s-module is essentialy embeddable in direct summand of s if and only if r is a direct sum of right uniserial rings if and only if r is a semiperfect ring whose cyclics are essentially embeddable in a direct summand of r.‎‏