Document Type: Research Paper
Department of Mathematical Sciences, University of Isfahan, Isfahan, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran.
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.