The Krum Dimension And Noether Dimension Of EN-modules And EA-modules

In this paper, through the essential submodules, we give two newmodules called EN-modules and EA-modules which like Noetherian modules andArtinian modules. Basing on studying the properties of these two modules, wegive these two modules’s structure, and the equivalent descriptions of some com-mon modules and rings. For example, the ring R is a Noetherian ring if thering R as a R-module is an EN-module. The ring R is an Artinian ring if thering R as a R-module is an EA-module, and the uniform dimension of R as aR-module is finite. The ring R is a semisimple ring if the ring R as a R-moduleis an EA-module and V-module. Lastly, as we can research Artinian modulesand EA-modules by Krull dimension, we quote a dimension called Noehteriandimension for researching EN-modules, and calculate their dimension.