Abstract
Discrete (quasi-discrete) modules form an important class in module theory, they are studied extensively by many authors. The class of lifting modules is obtained by considering only one of the defining conditions of quasi-discrete modules, namely the condition (D1). Here we focus on and study principally lifting modules, or modules with the condition (PD1). These modules are generalizations of lifting modules. We also study direct sums of P-hollow (semi-hollow) modules. It is known that relative projectivity is essential to study direct sums of quasi-discrete modules. Here we introduce the definition of relative Pprojectivity, which is essential to examine direct sums of hollow, and of P-hollow (semi hollow), modules for being principally lifting. Quasi-discrete module are always direct sums of hollow submodules, we show that finite dimensional modules with the condition (PD1) are direct sums of P-hollow (semi-hollow) submodules. We also obtain some properties for modules with (PD1), which are in analogy with the known properties for lifting modules.
Keywords
References
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- S. Singh, Semi-dual continuous modules over Dedekind domains , J. Univ. Kuwait Sci., 11 (1984), 33-39.
- R. Wisbauer, Foundations of module and ring theory, Gordon and Breach , Reading, (1991).
- Mahmoud A. Kamal * and Ahmed Yousef
- Faculty of Education, Ain Shams Univ., Roxy, Cairo, Egypt.
- E-mail: * mahmoudkamal333@hotmail.com
Abstract
References
- J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Brikh¨auser- Basel, 1st Edition, (2006).
- M.A. Kamal and A. Yousef, On supplementation and generalized projective modules, to appear in J. Egyptian Math. Soc..
- F. Kasch, Modules and rings, Academic press, (1982).
- S. Mohamed and F. Abdul-Karim, Semidual continuous abelian groups, J. Univ. Kuwait Sci., 11 (1984), 23-27.
- S. Mohamed and B. J. M¨uller, Continuous and discrete modules, Cambridge University Press (1990).
- S. Mohamed and B. J. M¨uller, Cojective modules, J. Egyptian Math. Soc., 12 (2004), 83-96.
- S. Mohamed and S. Singh, Generalizations of decomposition theorems known over perfect rings, J. Austral. Math. Soc., 24 (1977), 496-510.
- K. Oshiro, Semiperfect modules and quasi-semiperfect modules, Osaka J. Math., 20 (1983), 337-372.
- S. Singh, Semi-dual continuous modules over Dedekind domains , J. Univ. Kuwait Sci., 11 (1984), 33-39.
- R. Wisbauer, Foundations of module and ring theory, Gordon and Breach , Reading, (1991).
- Mahmoud A. Kamal * and Ahmed Yousef
- Faculty of Education, Ain Shams Univ., Roxy, Cairo, Egypt.
- E-mail: * mahmoudkamal333@hotmail.com
Details
| Other ID | JA82UJ55FN |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | December 1, 2007 |
| Published in Issue | Year 2007 Volume: 2 Issue: 2 |