A note on a generalization of injective modules - Türkmen, B.N.; Türkmen, E. (Carpathian Mathematical Publications, 2020)
As a proper generalization of injective modules in term of supplements, we say that a module M has the property (ME) if, whenever MN, M has a supplement K in N, where K has a mutual supplement in N. In this study, we obtain that (1) a semisimple R-module M has the property (E) if and only if M has the property (ME); (2) a semisimple left R-module M over a commutative Noetherian ring R has the property (ME) if and only if M is algebraically compact if and only if almost all isotopic components of M are zero; (3) a module M over a von Neumann regular ring has the property (ME) if and only if it is injective; (4) a principal ideal domain R is left perfect if every free left R-module has the property (ME)