Rickart and Dual Rickart Objects in Abelian Categories: Transfer via Functors

Abstract

Abstract. We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.

Strongly CS-Rickart and dual strongly CS-Rickart objects in abelian categories

“…In modules categories, they have been introduced and studied by Abyzov, Nhan and Quynh [1,2] and Tribak [34]. Rickart objects and their duals in abelian categories have been introduced and studied by Crivei, Kör and Olteanu [8,9], and subsume previous work of Dȃscȃlescu, Nȃstȃsescu, Tudorache and Dȃuş [13] on regular objects in abelian categories, Lee, Rizvi and Roman [24,25] on Rickart and dual Rickart modules, and in particular, Rizvi and Roman [31,32] and Keskin Tütüncü and Tribak [23] on Baer and dual Baer modules. The study of Baer modules, Rickart modules and their duals have the origin in the work of von Neumann on regular rings [35], Kaplansky [20] on Baer rings and Maeda [26] on Rickart rings.…”

Transfer of CS-Rickart and dual CS-Rickart properties via functors between abelian categories

“…But these dual results follow automatically by the duality principle in abelian categories, showing the advantage of working at this level of generality. In order to point out some previous interest in such topics, we mention Rickart and dual Rickart objects studied by Crivei, Kör and Olteanu [8,9], which have been proved to be useful in the study of regular objects in abelian categories in the sense of Dȃscȃlescu, Nȃstȃsescu, Tudorache and Dȃuş [12]. They generalize the moduletheoretic notions of Rickart and dual Rickart modules in the sense of Lee, Rizvi and Roman [19,20], and in particular, Baer and dual Baer modules studied by Rizvi and Roman [24,25] and Keskin Tütüncü and Tribak [17] respectively.…”

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

“…Thus, Lee, Rizvi and Roman introduced and extensively studied Rickart modules and dual Rickart modules in a series of papers [22,23,24], which extended the theory of Baer modules [31] and dual Baer modules [21]. A further generalization was considered by Crivei and Kör [5], who investigated relative Rickart objects and dual relative Rickart objects in abelian categories (also, see [6,7]). If M and N are objects of an abelian category, then N is called M -Rickart if for every morphism f : M → N , ker(f ) is a section, while N is called dual M -Rickart if for every morphism f : M → N , coker(f ) is a retraction.…”

“…If M and N are objects of an abelian category, then N is called M -Rickart if for every morphism f : M → N , ker(f ) is a section, while N is called dual M -Rickart if for every morphism f : M → N , coker(f ) is a retraction. 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.…”