On Injective Modules and Locally Nilpotent Endomorphisms of Injective Modules

First, injective modules are one of the most popular objects in homological algebra. In most cases, base rings are commutative and Noetherian so that the testing the injectivity of a given module is an important topic. Bear's criterion for injective modules over any ring gives a big tool to classify injective modules. Every morphism from an ideal I of R should be extended to the whole ring R to be an injective module R-module. In this paper, we can show that the Baer's test can be reduced from all ideals of R to all prime ideals of R to test the injectivity of a given R-module M if the base ring R is commutative and Noetherian. Second, the Enochs' Theorem can be extended to an arbitrary sequence {ƒi} of endomomorphisms of an injective left Noetherian and if a diagram of the minimal injective resolution of RM is commutative, then the locally nilpotence of ƒ implies the locally nilpotence of other maps in the diagram.