Let X be a class or R-modules containing 0 and closed under isomorphic images. With any such X we associate three classes ΓX, FX and ΔX. The study of some of the closure properties of these classes allows us to obtain characterization of Artinian modules dualizing results of Chatters. The theory of Dual Glodie dimension as developed by the author in some of his earlier work plays a crucial role in the present paper.
Let be a ring and an endomorphism of . We give a generalization of McCoy’s Theorem [ Annihilators in polynomial rings, Amer. Math. Monthly 64 (1957), 28–29] to the setting of skew polynomial rings of the form . As a consequence, we will show some results on semicommutative and -skew McCoy rings. Also, several relations among McCoyness, Nagata extensions and Armendariz rings and modules are studied.
A module M satisfies the restricted minimum condition if M/N is artinian for every essential submodule N of M. A ring R is called a right RM-ring whenever satisfies the restricted minimum condition as a right module. We give several structural necessary conditions for particular classes of RM-rings. Furthermore, a commutative ring R is proved to be an RM-ring if and only if R/Soc(R) is noetherian and every singular module is semiartinian.
In this article we introduce and study the concept of α-almost Artinian modules. We show that each α-almost Artinian module M is almost Artinian (i.e., every proper homomorphic image of M is Artinian), where α ∈ {0,1}. Using this concept we extend some of the basic results of almost Artinian modules to α-almost Artinian modules. Moreover we introduce and study the concept of α-Krull modules. We observe that if M is an α-Krull module then the Krull dimension of M is either α or α + 1.