On generalized submodules of QTAG-modules

Abstract

If α denotes the class of all QTAG-modules M such that M/H β (M) is totally projective for every ordinal β < α, then these modules are called α-modules. Here we study the relation between the structure of fully invariant submodules of certain QTAG-modules and the structure of containing modules. It is found that if F is a fully invariant submodule of the totally projective QTAG-module M, then both F and M/F are totally projective. We show that if for some sequence β = (β k ) k<ω , both M β and M/M β are totally projective, then M itself is necessarily totally projective.

Notes on summability in $QTAG$-modules

“…(For the corresponding results concerning totally projective modules we refer once again to [1]). In some instance this is so; for examples:…”

“…Let β = {β k } k<ω be an increasing sequence of ordinals and symbols ∞; that is, for each k, either β k is an ordinal or β k = ∞ and β k < β k+1 provided β k = ∞. Imitating [1], with each such sequence β we associate the fully invariant submodule M β of the QT AG-module M as…”

“…In view of [1], we obtain that M is σ-summable. Thus H ω (M ) is σ-summable, i.e., a direct sum of uniserial modules and hence summable.…”

“…Firstly, if length of M = λ < µ, we get that M is a direct sum of countably generated modules. Hence, [1]'s result is applicable to deduce that L is totally projective of countable length, that is a direct sum of countably generated modules, whence summable. Otherwise, if length of M < λ, M is obviously a direct sum of countably generated modules and again the previous procedure can be applied.…”

“…Recall that a submodule F of a module M is said to be fully invariant [8] if each endomorphism of M sends F into itself. In [1], the first author proved that if F is a fully invariant submodule of the totally projective module M , then both F and M/F are totally projective modules; and, conversely, if both F and M/F are totally projective, then M is itself totally projective, where the fully invariant submodule F of M is of concrete type. The general case for F is still unanswered.…”