A ring R is called right almost MGP-injective (or AMGP-injective for short) if, for any <$E0~symbol Щ~a~symbol <174>~R>, there exists an element <$Eb~symbol <174>~R> such that <$Eab~=~ba~symbol Щ~0> and any right R-monomorphism from abR to R extends to an endomorphism of R. In this paper, several properties of these rings are given, some interesting results are obtained. Using the concept of right AMGP-injective rings, we present some new characterizations of QF-rings, semisimple Artinian rings and simple Artinian rings.