: In t his Thesis, the properties of almost injective modules is investigated. We provide a negative answer to a question raised by Jain and Alahmadi asking for a possible “Baer-like” criterion for almost injective modules. It is shown that there exists an R-module M which is almost injective relative to R but not almost injective. Also, we study the rings R for which every R -module is almost injective. For such a ring R , it is shown that R/ Soc(R) is semisimple and Rad(R) is fnitely generated. It is proved that these rings are exactly Artinian serial rings with Rad(R)^2 = 0 , if one of these conditions hold: Soc(R) is fnitely generated, R is right extending, R is semiperfect or R is of fnite reduced rank. Furthermore, we introduce the normal; MARGIN: 0cm 0cm 0pt; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt" 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. Key Words: Almost injective modules, Right V -rings, Right almost V -rings, Essentially injective modules, simple-extending modules.
