Rickart and Dual Rickart Objects in Abelian Categories

Abstract

We introduce and study relative Rickart objects and dual relative Rickart objects in abelian categories. We show how our theory may be employed in order to study relative regular objects and (dual) relative Baer objects in abelian categories. We also give applications to module and comodule categories.

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

“…Now let us recall from [6,9] the definitions of the main concepts of the present paper. (b) Let us point out the difference between the terminology used in the theory of relative regular objects in [9] and continued by us in [6] and in the present paper, and the terminology used in the theory of (dual) relative Rickart modules in [18,19]. In the latter the roles of M and N are swapped, so that a module M is called N -Rickart if and only if the kernel of every morphism f : M → N is a section, and dual N -Rickart if and only if the cokernel of every morphism f : M → N is a retraction.…”

“…When A is an abelian category, N is M -regular if and only if for every morphism f : M → N , Ker(f ) is a direct summand of M and Im(f ) is a direct summand of N [9, Proposition 3.1]. The main idea from [6] was to split the study of these two conditions characterizing relative regularity. Thus, N is called M -Rickart if for every morphism f : M → N , Ker(f ) is a direct summand of M ; also, N is called dual M -Rickart if for every morphism f : M → N , Im(f ) is a direct summand of N .…”

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].…”

“…The concepts of extending and lifting modules are well established in Module Theory [7,19]. The interest in (dual) relative CS-Rickart objects stems, on one hand, from the module-theoretic work of Lee, Rizvi and Roman [25,26], and on the other hand, from their applications to relative regular objects in abelian categories studied by Dȃscȃlescu, Nȃstȃsescu, Tudorache and Dȃuş [18], as shown by Crivei and Kör [11]. The setting of abelian categories has the advantage that one may freely use the duality principle in order to automatically obtain dual results, and yields applications to categories other than module categories.…”

“…The setting of abelian categories has the advantage that one may freely use the duality principle in order to automatically obtain dual results, and yields applications to categories other than module categories. Furthermore, (dual) relative Rickart objects generalize (dual) relative Baer objects, which were studied by Rizvi and Roman [31,32], Keskin Tütüncü and Tribak [23] in module categories, and Crivei and Kör [11] in abelian categories. Likewise, as a relevant subclass of (dual) relative CS-Rickart objects, in the present paper we will consider and investigate (dual) relative CS-Baer objects in abelian categories.…”