CS-Rickart modules

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

“…Let A be an abelian category. (1) Recall that a Grothendieck category A is called a V -category if every simple object is injective [14], and a regular category if every object B of A is regular in the sense that every short exact sequence of the form 0…”

“…(Dual) CS-Rickart objects in abelian categories have been introduced by the authors in [12] as a common generalization of (dual) Rickart objects and extending (lifting) objects. 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.…”

“…Now we have the following specializations of the above notions by restricting to fully invariant sections and fully coinvariant retractions. Such notions have already been considered in module categories[1,2,16,36]. Rickart if for every morphism f : M → N there are a fully coinvariant retraction r : N → P and a superfluous epimorphism t : P → Coker(f ) in A such that coker(f ) = tr.…”

CS-Baer and dual CS-Baer objects in abelian categories

“…Let M and N be modules over a unitary ring. Then N is called M-CS-Rickart if the kernel of every homomorphism f : M → N is essential in a direct summand of M, and dual M-CS-Rickart if the image of every homomorphism f : M → N lies above a direct summand of N in the sense that Im(f )/L is superfluous in M/L for some direct summand L of M. Inspired by the work of Abyzov and Nhan on (dual) CS-Rickart modules [1,2], we have considered and studied (dual) relative CS-Rickart objects in abelian categories [14,15], as a generalization of both extending (lifting) and (dual) relative Rickart objects [11]. The concepts of extending and lifting modules are well established in Module Theory [7,19].…”

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

“…Extending modules (also called CS-modules) and lifting modules have been intensively investigated for the last decades, due to their important applications to ring and module theory (e.g., see [5,13]). Let us also note that CS-Rickart objects and dual CS-Rickart objects extend the module-theoretic concepts of CS-Rickart and dual CS-Rickart modules studied by Abyzov, Nhan and Quynh [1,2]. In our approach we needed to develop specific categorical techniques in order to generalize results on (dual) CS-Rickart modules and extending (lifting) modules to abelian categories.…”