A ring R is called right AGP-injective if, for any 0 = a ∈ R, there exists n > 0 such that a~n = 0 and Ra~n is a direct summand of lr(a~n). In this paper some conditions which are sufficient or equivalent for a right AGP-injective ring to be von Neumann regular (right self-injective, semisimple) are provided. It is shown that a ring R is von Neumann regular if and only if R is right AGP-injective and for any 0 = a ∈ R there exists a positive integer n with 0 = a~n such that a~n R is a projective right R-module if and only if R is a right AGP-injective ring whose divisible and torsionfree right ii-modules are GP-injective. We also show that if R is a primitively finite right AGP-injective ring, then R approx= R_1 * R_2, where R_1 is semisimple and every simple right ideal of R_2 is nilpotent. In addition, it is proven that if R is a right MI and right AGP-injective ring satisfying the a.c.c on right annihilators, then R is quasi-Frobenius.
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