Direct sums of injective and projective modules

It is well known that a countably injective module is Σ-injective. In [K.I. Beidar, S.K. Jain, Ashish K. Srivastava, New characterization of Σ-injective modules, Proc. Amer. Math. Soc. 316 (10) (2008) 3461–3466], Beidar, Jain and Srivastava extended it and showed that an injective module M is Σ-injective if and only if each essential extension of M(ℵ0) is a direct sum of injective modules. This paper extends and simplifies this result further and shows that an injective module M is Σ-injective if and only if each essential extension of M(ℵ0) is a direct sum of modules that are either injective or projective. Some consequences and generalizations are also obtained.