Rings for which every simple module is almost injective

‎We introduce the class of ``right almost $V$-ringsʹʹ 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 $2 \times 2$ upper triangular matrix ring over $R$ is a right almost $V$-ring precisely when $R$ is semisimple‎. ‎‎