Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X42120160201Rings for which every simple module is almost injective113127747ENSh.AsgariDepartment of Mathematical Sciences, University of Isfahan, Isfahan, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.M.Arabi-KakavandDepartment of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran.H.KhabazianDepartment of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran.0000-0003-2091-6940Journal Article20140817We introduce the class of “right almost V-rings” which is properly between the <br />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 <br />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.http://bims.iranjournals.ir/article_747_f6b734ed6ae927135539c5e60f93a8b0.pdf