ON ALMOST WN-INJECTIVE RINGS

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 N(R) ,there exists n ≥1 and an S-submodule a X_a of M such that an a^n #0 and l_m(r_R(a^n))= Ma^n + X_a^n 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 .