A lifting theorem for modular representations

Journal of Al-Qadisiyah for Computer Science and Mathematics

In this work we will attempt to define and investigate new classes of modules named -g-supplemented and -g-radical supplemented as a proper generalization of class of g-lifting modules and identify several distinct characterizations of these modules. Additionally, we'll attempt to explain the concepts of projective g-covers and g-semiperfect modules. It is shown that the two buildings of g-semiperfect and -g-supplemented modules are the same for the class of projective modules.

Rocky Mountain Journal of Mathematics, 2017

Let R be an arbitrary ring with identity and M a right R-module. In this paper, we introduce a class of modules which is analogous to that of Goldie *-lifting and principally Goldie *-lifting modules. The module M is called principally G *-δ-lifting if, for any m ∈ M , there exists a direct summand N of M such that mR is β * δequivalent to N. We also introduce a generalization of Goldie *-supplemented modules, namely, a module M is said to be principally G *-δ-supplemented if, for any m ∈ M , there exists a δ-supplement N in M such that mR is β * δ-equivalent to N. We prove that some results of principally G *-lifting modules and Goldie *-lifting modules can be extended to principally G *-δ-lifting modules for this general setting. Several properties of these modules are given, and it is shown that the class of principally G *-δ-lifting modules lies between the classes of principally δ-lifting modules and principally G *-δ-supplemented modules. 2010 AMS Mathematics subject classification. Primary 16D10, 16D70, 16D80. Keywords and phrases. δ-small submodule, δ-supplement, β * δ relation, principally G *-δ-lifting module, principally G *-δ-supplemented module.

Proceedings of The London Mathematical Society, 1991

2010

The notion of projection invariant subgroups was first introduced by Fuchs in (7). We will define the module-theoretic version of the projection invariant subgroup. Let R be a ring and M a right R-module. We call a submodule N of M the projection invariant if every projectionof M onto a direct summand maps N into itself, i.e. N is invariant under any projection of M. In this note, we give several characterizations to these class of modules that generalize the recent results in (14). We also define and study the PI-lifting modules which is a generalization of FI-lifting module. It is shown that if each Mi is a PI-lifting module for all 1 ≤ i ≤ n, then M = ⊕ n=1Mi is a PI-lifting module. In particular, we focus on rings satisfying the following condition: (*) Every submodule of M is projection invariant. We prove that if R has the (∗) property, then R ⊕ R does not satisfy the (∗) property.

In this paper, we introduce principally δ-lifting modules which are analogous to δ-lifting modules and principally δ-semiperfect modules as a generalization of δ-semiperfect modules and investigate their properties. 2000 Mathematics Subject Classification. 16U80.

Journal of Algebra, 2007

Let F and K be algebraically closed fields of characteristics p > 0 and 0, respectively. For any finite group G we denote by KR F (G) = K ⊗ Z G 0 (FG) the modular representation algebra of G over K where G 0 (FG) is the Grothendieck group of finitely generated FG-modules with respect to exact sequences. The usual operations induction, inflation, restriction, and transport of structure with a group isomorphism between the finitely generated modules of group algebras over F induce maps between modular representation algebras making KR F an inflation functor. We show that the composition factors of KR F are precisely the simple inflation functors S i C,V where C ranges over all nonisomorphic cyclic p-groups and V ranges over all nonisomorphic simple K Out(C)-modules. Moreover each composition factor has multiplicity 1. We also give a filtration of KR F .

Vietnam Journal of Mathematics, 2013

Following (Kosan in Algebra Colloq. 14:53-60, 2007), a module M is called δsupplemented if, for every submodule of N of M, there exists L ≤ N such that M = N + L and N ∩L δ L. A module M is called δ-lifting if, for any N ≤ M, there exists a decomposition M = A ⊕ B such that A ≤ N and N ∩ B δ M. In this paper, we study e-supplemented modules and e-lifting modules which are generalized δ-supplemented modules and δ-lifting modules. Some properties of these modules are considered. Moreover, we also have new characterizations of Artinian (resp., Noetherian) Rad e (M) module studied with chain conditions on e-supplemented modules.

Promoter: Professor N. J. Groenewald 0.1. ABSTRACT i 0.1 Abstract This thesis is aimed at generalizing notions of rings to modules. In particular, notions of completely prime ideals, s-prime ideals, 2-primal rings and nilpotency of elements of rings are respectively generalized to completely prime submodules and classical completely prime submodules, s-prime submodules, 2-primal modules and nilpotency of elements of modules. Properties and radicals that arise from each of these notions are studied. ii 0.2 Acknowledgement Firstly, I express gratitude to my promoter Prof. N. J. Groenewald. His invaluable assistance, guidance and advice which was more than just academic led me this far. I am indebted to DAAD, NRF and NMMU for the financial support. I thank Prof. Straeuli, Prof. Booth, Ms Esterhuizen and all staff of the Department of Mathematics and Applied Mathematics of NMMU for being hospitable and for providing an environment conducive for learning -baie dankie! I owe gratitude to Prof. G. K.

Journal of Algebra and Its Applications, 2019

The paper is focused on questions when some homological and submodule-chain conditions satisfied by a module [Formula: see text] are preserved by the group module [Formula: see text]. Namely, it is proved for a group [Formula: see text] and an [Formula: see text]-module [Formula: see text] that [Formula: see text] is flat if and only if [Formula: see text] is flat, and [Formula: see text] is artinian if and only if [Formula: see text] is artinian and [Formula: see text] is finite, which are two questions raised by Yiqiang Zhou: On Modules Over Group Rings, Noncommutative Rings and Their Applications LENS July 1-4, 2013.