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.