JZS (2015) 17- 4 (Part-A) 51               Abdullah M. Abdul-Jabbar Department of Mathematics, College of Science, Salahaddin University-Erbil, Kurdistan Region Iraq E-mail:abdullah.abduljabbar@su.edu.krd Article info Abstract Original: 27 Mar 2015 Revised: 1 May 2015 Accepted: 31 May 2015 Published online: 20 Dec. 2015 A ring R is called left AGP-injective if for any 0 ≠ a ∈ R, there exists a positive integer n such that a n ≠ 0 and a n R is a direct summand of rℓ(a n ). Now, in the present paper, we invistigate some properties of rings whose simple singular right R-modules are AGP- injective. Also, we give a characterization of π-regular rings interms of right weakly continuous ring whose simple singular right R-modules are AGP-injective under the condition, MERT ring. Finally, we give a property of AGP-injective rings with an index set {Xa n : a ∈ R and n is a positive integer} of ideals such that Xa n b = Xba n , for all a, b ∈ R and a positive integer n. Key Words: AP-injective modules, AGP-injective modules, AGP-injective rings, regular rings. 1. Introduction and Preliminaries Throughout this paper, R denotes an associative ring with identity and all modules are unitary. For a subset X of R, the right annihilator of X in a ring R is defined by r(X) = {t∈R: xt = 0, for all x∈X}. Similarly, define the left annihilator of X in a ring R as ℓ(X) = { t∈R: tx = 0, for all x∈X}. If X = {a}, we usually us to the abbreviation r(a) (resp. ℓ(a)). An ideal I of a ring R is said to be essential if and only if I has a non-zero intersection with every non-zero ideal of R. Let R be a ring and x an element in R. Then x is said to be a right singular if and only if r(x) is an essential right ideal of R. The set of all right sigular elements in R is denoted by Y(R). Y(R) is a right ideal of R, which is called the right singular ideal of R. The left singular ideal Z(R) is similarly defined. A ring R is called right (left) non-singular if Y(R) = (0) (resp. Z(R) = (0)). For a left R-module M, Z(M) = {z ∈ M: ℓ(z) is an essential left ideal of R} is called the left singular submodule of M. M is called left non-singular (resp. singular) if Z(M) = (0) (resp. Z(M) = M). The Jacobson radical [3] of a ring R, denoted by J(R), is the intersection of all maximal Journal homepage www.jzs.uos.edu.krd Journal of Zankoi Sulaimani Part-A- (Pure and Applied Sciences)