On $\mathcal{F}$-cosmall morphisms

Abstract

In this paper, we first define the notion of $\mathcal{F}$-cosmall quotient for an additive exact substructure $\mathcal{F}$ of an exact structure $\mathcal{E}$ in an additive category $\mathcal{A}$. We show that every $\mathcal{F}$-cosmall quotient is right minimal in some cases. We also give the definition of $\mathcal{F}$-superfluous quotient and we relate it the approximation of modules. As an application, we investigate our results in a pure-exact substructure $\mathcal{F}$.

Finitely-cosmall Quotients

“…In [9], the author dualized some results of Cortés-Izurdiaga in [10] and got several useful results by investigating the relationship between 𝐸𝑛𝑑 𝑅 (𝑁) and 𝐸𝑛𝑑 𝑅 (𝑀) when there is a right minimal epimorphism 𝑝: 𝑀 → 𝑁. In [7], Kaleboğaz and Keskin Theorem 2.14 Let 𝑃 be a right 𝑅-module which is projective with respect to a finitely (singly) split epimorphisms. Every finitelycosmall quotient (singly-cosmall quotient) 𝑓: 𝑃 → 𝑀 is right minimal.…”

“…Then, in [7], Kaleboğaz and Keskin Tütüncü first introduced F-cosmall quotient morphisms with respect to an additive exact substructure F of an exact structure in an additive category by using F-copartial morphisms and they gave an application of this definition to a pure-exact substructure F in the category of right modules over a ring and they called that kind of morphisms purecosmall quotients. In this paper, we first give the definiton of finitely-cosmall quotients (singly-cosmall quotients) as an another application of F-cosmall quotients to the finite (single) pure-exact substructure F in the category of right modules over a ring (see in Definition 2.9).…”

“…In [9], right minimal morphisms were defined by Keskin Tütüncü as a dual definition of left minimal morphisms which were studied by Cortés-Izurdiaga et al in [10]. In [7], Kaleboğaz and Keskin Tütüncü gave an example of right minimal morphisms by using pure-cosmall quotients.…”

“…Pure-cosmall quotient morphisms are first defined by Kaleboğaz and Keskin Tütüncü[7, Definition 4], as an application of Fcosmall morphisms to a pure-exact substructure F in the category of right 𝑅modules over a ring 𝑅 as follows: Definition 2.8 Let 𝑌, 𝑈 be right 𝑅-modules and 𝑝: 𝑌 → 𝑈 be an epimorphism. 𝑌 is purecosmall in 𝑈 if for any copartial morphism 𝑔: 𝑍 → 𝑌 from a right 𝑅-module 𝑍 to 𝑌 with codomain 𝑌, the following holds:…”