In this article we introduce and study the concept of˛-almost Artinian modules. We show that each -almost Artinian module M is almost Artinian (i.e., every proper homomorphic image of M is Artinian), wherę 2 f0; 1g. Using this concept we extend some of the basic results of almost Artinian modules to˛-almost Artinian modules. Moreover we introduce and study the concept of˛-Krull modules. We observe that if M is an˛-Krull module then the Krull dimension of M is either˛or˛C 1.

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