Abstract
A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. The study of large submodules and its fascinating properties makes the theory of QTAG-modules more interesting. A fully invariant submodule L of M is large in M if L+B=M, for every basic submodule B of M. The impetus of these efforts lies in the fact that the rings are almost restriction-free. This motivates us to find the necessary and sufficient conditions for a submodule of a QTAG-module to be large and characterize them. Also, we investigate some properties of large submodules shared by Σ-modules, summable modules, σ-summable modules, and so on.
Highlights
Introduction and PreliminariesAll the rings R considered here are associative with unity and modules M are unital QTAG-modules
We investigate some properties of large submodules shared by Σ-modules, summable modules, σ-summable modules, and so on
M is h-divisible if M = M1 = ⋂∞ k=0 Hk(M) and it is h-reduced if it does not contain any h-divisible submodule
Summary
All the rings R considered here are associative with unity and modules M are unital QTAG-modules. An element x ∈ M is uniform, if xR is a nonzero uniform ( uniserial) module and, for any R-module M with a unique composition series, d(M) denotes its composition length. M is h-divisible if M = M1 = ⋂∞ k=0 Hk(M) and it is h-reduced if it does not contain any h-divisible submodule In other words it is free from the elements of infinite height. A QTAG-module M is called σ-summable if Soc(M) = ⋃n
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