On Almost WN-Injective Rings

Abstract

Let R be a ring. Let  M_R be a module with S=End (M_R). The module ^M is called almost Wnil-injective (briefly right AWN-injective ) if, for any 0 ≠ a є N(R), there exists n≥1 and an S-submodule  X_a of M such that n^a ≠ 0 and l_m(r_R(aⁿ)) = Maⁿ ⊕ X_an as left S-modules .If R_R is almost Wnil-injective , then we call R is right almost Wnil-injective ring . In this paper ,we give some characterization and properties of almost Wnil-injective rings .In particular , Conditions under which right almost Wnil-injective rings are n-regular rings and n-weakly regular rings are given .Also we study rings whose simple singular right R-module are almost Wnil-injective , It is proved that if R is a NCI ring ,MC2 , whose every simple singular R-module is almost Wnil-injective , Then R is reduced.

Keywords: Nil-Injective, Reduced Ring , MC2 Ring , N-Regular Ring