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Almost N-projective modules and generalized N-projective modules play an important role in the study of lifting modules and artinian rings. In this paper, we study a relationship between almost N-projective modules and generalized N-projective modules, and give new characterizations of almost N-projective modules and generalized N-projective modules by homomorphisms between their projective covers, respectively. Moreover, using the result, we consider a condition for a module which is almost N i -projective for any i ∈ I to be almost ⊕ i ∈ I N i -projective. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concept of "factor-square-full" modules defined as follows: For any proper submodule X of M, there exist a proper submodule Y of M with X ⊆ Y and an epimorphism f : M → (M / Y) 2 , and give some properties of these modules. Using those results, we show that any quasi-discrete module is a direct sum of a factor-square-full module and a dual-square-free module which are dual orthogonal, and consider the problem "When is a quasi-discrete module quasi-projective?". Moreover, we study rings whose factor-square-full modules are closed under essential extensions or essential submodules. [ABSTRACT FROM AUTHOR]

A module M is said to be square free if whenever its submodule is isomorphic to N2 = N⊕N for some module N, then N = 0. Dually, a module M is said to be d-square free (dual square free) if whenever its factor module is isomorphic to N2 for some module N, then N = 0. In this paper, we give some fundamental properties of d-square free modules and study rings whose d-square free modules are closed under submodules or essential extensions. [ABSTRACT FROM AUTHOR]

In 1971, Koehler [11] proved a structure theorem for quasi-projective modules over right perfect rings by using results of Wu-Jans [22]. Later Mohamed-Singh [17] studied discrete modules over right perfect rings and gave decomposition theorems for these modules. Moreover, Oshiro [18] deeply studied (quasi-)discrete modules over general rings. In this paper, we consider that decomposition theorems for H-supplemented modules with the condition (D2) or (D3) over right perfect rings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a direct projectiveH-supplemented module which is a generalization of a discrete module (cf. [11, 16]). We prove that any direct projectiveH-supplemented module satisfies the exchange property and also shows that ifM1andM2are direct projectiveH-supplemented modules, thenM = M1⊕M2is direct projectiveH-supplemented if and only ifMiis radicalMj-projective (i ≠ j). [ABSTRACT FROM PUBLISHER]

A moduleMis said to becontinuousif it is extending with the condition (C2) (cf. [6], [7]). In this article, we consider a 𝒢-extending module with (C2) which is a generalization of a continuous module. First, we show that any 𝒢-extending module with (C2) satisfies the exchange property. We also prove that, ifM1andM2are 𝒢-extending modules with (C2), thenM1⊕M2is 𝒢-extending with (C2) if and only ifMiisMj-ejective (i ≠ j). [ABSTRACT FROM PUBLISHER]

In this article, we introduce a generalization of quasi-discrete (a GQD-module) by using the notion ofH-supplemented modules and investigate some properties of GQD-modules. First we consider some properties of a relative radical projectivity which is useful in analyzing the structure ofH-supplemented modules. We apply them to the study of direct sums of GQD-modules. Moreover, we prove that anyH-supplemented (lifting) module with finite internal exchange properly (FIEP) has an indecomposable decomposition and show that, for anH-supplemented (lifting) module, the finite exchange property implies the full exchange property. [ABSTRACT FROM PUBLISHER]

Harada (cf. 1983) introduced the lifting property for maximal submodules and the extending property for simple submodules, and Oshiro (cf. 1983b) definitely introduced lifting modules and extending modules. Since then, the method and results in their works urged not only the study on (quasi-) discrete and (quasi-) continuous modules but also the one on themselves. The reader is referred to Mohamed and Müller (1990) and Dung et al. (1994) for the research on lifting modules and extending modules. In this article, we study the following fundamental open problems: When is a direct sum ⊕IMi of lifting (extending) modules {Mi}I lifting (extending)? These problems are unsolved even in the case that the index set I is finite. Now, the purpose of this article is to study the problem concerned with finite direct sums of lifting modules. [ABSTRACT FROM AUTHOR]