Direct injective modules were defined by W. K. Nicholson in 1976. An \(R\)- module \(M\) is said to be direct injective if given a direct summand \(N\) of \(M\) with inclusion \(i: N\to M\) and any monomorphism \(g: N\to M\), there exists an endomorphism \(f\) of \(M\) such that \(fg=i\). Properties of direct injective modules are investigated in the first part of the paper and a number of new characterizations are proved. An \(R\)-module \(M\) is called an extending module if every submodule \(N\) of \(M\) is essentially extended to a direct summand \(N^*\) of \(M\). The authors show that the endomorphism ring of an extending direct injective module \(M\) is a semiregular ring. This result is used to answer affirmatively the following question posed by Matlis in 1958: If \(M\) is a direct sum of indecomposable injective modules \(\{M_ i\}_ I\), is any summand of \(M\) a direct sum of indecomposable injective modules?