α-modules and generalized submodules

Rafiquddin 1 , Ayazul Hasan 2 , and Mohammad Fareed Ahmad 3
  • 1 Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, 202002, Aligarh, India
  • 2 College of Applied Industrial Technology, Jazan University, Saudi Arabia
  • 3 Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, 202002, Aligarh, India

Abstract

A QTAG-module M is an α-module, where α is a limit ordinal, if M/Hβ (M) is totally projective for every ordinal β < α. In the present paper α-modules are studied with the help of α-pure submodules, α-basic submodules, and α-large submodules. It is found that an α-closed α-module is an α-injective. For any ordinal ωαω1 we prove that an α-large submodule L of an ω1-module M is summable if and only if M is summable.

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