Rings for which every simple module is almost injective

We introduce the class of “right almost V-rings” which is properly between the
classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if
for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-ring if and only if for every simple R-module S, either S is injective or the injective hull of S is projective of length 2. Right Artinian right almost V-rings and right Noetherian right almost V-rings are characterized. A 2×2 upper triangular matrix ring over R is a right almost V-ring precisely when R is semisimple.