Modules in Which Inverse Images of Some Submodules are Direct Summands

Split objects with respect to a fully invariant short exact sequence in abelian categories

“…The next result generalizes [35,Proposition 2.16], without the quasi-projectivity condition. (1) M is (strongly) self-F -split if and only if for every subobject…”

“…Let M and N be right R-modules. Then N is M -F -split if and only if N is M -F -inverse split in the sense of [35]. For F = Z 2 M (N ) (see the notation from the last section of our paper), a module N is M -F -split if and only if N is M -T -Rickart in the sense of [14].…”

“…Their motivation was to set a unified theory of relative Rickart objects with versatile applications, which allows one to deduce naturally properties of dual relative Rickart objects (by the duality principle), relative regular objects (which are relative Rickart and dual relative Rickart) in the sense of Dȃscȃlescu, Nȃstȃsescu, Tudorache and Dȃuş [11,12] as well as relative Baer objects (as particular relative Rickart objects) and dual relative Baer objects (by the duality principle) [5,6,7]. In recent years, Rickart modules and dual Rickart modules were generalized to (dual) t-Rickart modules by Asgari and Haghany [1], (dual) T -Rickart modules by Ebrahimi Atani, Khoramdel and Dolati Pish Hesari [14,16] and (dual) F -inverse split modules by Ungor, Halicioglu and Harmanci [35,36], the latter developing a theory using arbitrary fully invariant submodules.…”

Module decompositions by images of fully invariant submodules

“…is a direct summand of M for every f ∈ S. On the other hand, in [15], a module M is said to be F-inverse split if f −1 (F) is a direct summand of M for every f ∈ S. There are some interesting connections between these classes of modules. For example, in [15], it is proved that M is F-inverse split if and only if M has a decomposition M = F ⊕ N where N is a Rickart module. Since the second singular submodule Z 2 (M) of M is fully invariant in M, being a T-Rickart module and being a Z 2 (M)-inverse split module are the same.…”

“…Since the second singular submodule Z 2 (M) of M is fully invariant in M, being a T-Rickart module and being a Z 2 (M)-inverse split module are the same. Some applications of the notion of an F-inverse split module M are presented in [4], [14], [15] and [16] by considering certain fully invariant submodules aside from the second singular submodule.…”