As in earlier papers, the author introduces a generalization of the concept of injectivity and applies this to some classes of rings: A left A-module M is called E-injective if, for any complement left submodule C of M and relative complement K of C, any essential left submodule E of M containing \(K\oplus C\), any left A-monomorphism \(g: E\to M\) and A- homomorphism \(f: E\to M\), there exists an endomorphism h of M such that \(hg=f\). Some typical results involving this definition are: The ring A is semi-simple Artinian iff A is a semiprime left E-injective \(\Sigma\)-ring. The ring A is left and right continuous and strongly regular iff A is a reduced left E-injective ring. If A is a commutative E-injective ring with Jacobson radical J and with nilpotent singular ideal, then two injective modules M and N are isomorphic iff the annihilators \(r_ M(J)\) and \(r_ N(J)\) are isomorphic. There is also a characterization of strongly regular rings.