Let be an associative ring and be a left -module. We introduce the concept of the incidence module of a locally finite partially ordered set over . We study the properties of and give the necessary and sufficient conditions for the incidence module to be an IN-module, -module, nil injective module and nonsingular module, respectively. Furthermore, we show that the class of -modules is closed under direct product and upper triangular matrix modules.
We give some sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, normal EP element and strongly EP element by using solutions of certain equations.
We characterize semiperfect modules, semiperfect rings, and perfect rings using locally projective covers and generalized locally projective covers, where locally projective modules were introduced by Zimmermann-Huisgen and generalized locally projective covers are adapted from Azumaya’s generalized projective covers.
The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.
An interesting topic in the ring theory is the classification of finite rings. Although rings of certain orders have already been classified, a full description of all rings of a given order remains unknown. The purpose of this paper is to classify all finite rings (up to isomorphism) of a given order. In doing so, we introduce a new concept of quasi basis for certain type of modules, which is a useful computational tool for dealing with finite rings. Then, using this concept, we give structure...
Let be the -dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. We give the classification of all ideals of -dimensional Radford Hopf algebra by generators.
We classify the affine varieties of dimension at most 4 which occur as orbit closures with an invariant point in varieties of representations of quivers. Moreover, we show that they are normal and Cohen-Macaulay.
Let be an associative ring with identity and the Jacobson radical of . Suppose that is a fixed positive integer and an -torsion-free ring with . In the present paper, it is shown that is commutative if satisfies both the conditions (i) for all and (ii) , for all . This result is also valid if (ii) is replaced by (ii)’ , for all . Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).