For two modules M and N, (i)(M)(N) stands for the largest submodule of N relative to which M is injective. For any module M, (i)(M) : Mod-R -> Mod-R thus defines a left exact preradical, and (i)(M)(M) is quasi-injective. Classes of ring including strongly prime, semi-Artinian rings and those with no middle class are characterized using this functor: a ring R is semi-simple or right strongly prime if and only if for any right R-module M, (i)(M)(R) - R or 0, extending a result of Rubin; R is a right QI-ring if and only if R has the ascending chain condition (a.c.c.) on essential right ideals and (i)(M) is a radical for each M is an element of Mod-R (the a.c.c. is not redundant), extending a partial answer of Dauns and Zhou to a long-standing open problem. Also discussed are rings close to those with no middle class.