Rings all of whose right ideals are U-modules

The notion of a U-module was introduced and thoroughly investigated in [11 Ibrahim, Y., Yousif, M. F. (2017). U-modules. Comm. Algebra, doi:https://doi.org/10.1080/00927872.2017.1339064.[Crossref] , [Google Scholar]] as a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper a right R-module M is called a U*-module if every submodule of M is a U-module, and a ring R is called a right U*-ring if R_R is a U*-module. We show that M is a U*-module iff whenever A and B are submodules of M with A?B and A∩B = 0, A⊕B is a semisimple summand of M; equivalently M = X⊕Y, where X is semisimple, Y is square-free, and X &; Y are orthogonal. In particular, a ring R is a right U*-ring iff R is a direct product of a square-full semisimple artinian ring and a right square-free ring. Moreover, right U*-rings are shown to be directly-finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable-range 1, and hence is stably-finite. These results are non-trivial extensions of similar ones on rings all of whose right ideals are either quasi-continuous or auto-invariant.     

Simple-Direct-Projective Modules

In this paper, we introduce and study the dual notion of simple-direct-injective modules. Namely, a right R-module M is called simple-direct-projective if, whenever A and B are submodules of M with B simple and M/A ? B ?^⊕M, then A ?^⊕M. Several characterizations of simple-direct-projective modules are provided and used to describe some well-known classes of rings. For example, it is shown that a ring R is artinian and serial with J²(R) = 0 if and only if every simple-direct-projective right R-module is quasi-projective if and only if every simple-direct-projective right R -module is a D3-module. It is also shown that a ring R is uniserial with J²(R) = 0 if and only if every simple-direct-projective right R-module is a C3-module if and only if every simple-direct-injective right R -module is a D3-module.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

On ADS Modules and Rings

A right module M over a ring R is said to be ADS if for every decomposition M = S ⊕ T and every complement T′ of S, we have M = S ⊕ T′. In this article, we study and provide several new characterizations of this new class of modules. We prove that M is semisimple if and only if every module in σ[M] is ADS. SC and SI rings also characterized by the ADS notion. A ring R is right SC-ring if and only if every 2-generated singular R-module is ADS.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

Simple-direct-modules

A right R-module M is called simple-direct-injective if, whenever, A and B are simple submodules of M with A?B, and B?^⊕M, then A?^⊕M. Dually, M is called simple-direct-projective if, whenever, A and B are submodules of M with M∕A?B?^⊕M and B simple, then A?^⊕M. In this paper, we continue our investigation of these classes of modules strengthening many of the established results on the subject. For example, we show that a ring R is uniserial (artinian serial) with J²(R) = 0 iff every simple-direct-projective right R-module is an SSP-module (SIP-module) iff every simple-direct-injective right R-module is an SIP-module (SSP-module).     

Rings whose cyclic modules are lifting and ⊕-supplemented

It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules.