$${\mathcal {F}}$$-Copartial Morphisms

Finitely-cosmall Quotients

“…In [4,Proposition 3.8], Kaleboğaz proved that over a right Noetherian ring every copartial morphism is finitely and also singly copartial morphism. We can extend this result to the copartial isomorphisms and and finitely copartial isomorphisms as in the following: Corollary 2.12 Let 𝑅 be a ring.…”

“…And she called them finitely (singly) copartial morphisms (see in Definition 2.2). In [4], Kaleboğaz investigated the relations between copartial morphisms and finitely (singly) copartial morphisms. Moreover, she gave new characterizations of finitely (singly) pure projective modules, flat modules and finitely (singly) projective modules by using copartial morphisms and finitely (singly) copartial morphisms.…”

“…Then in [5, Theorem 3], Azumaya also proved that 𝑓 is finitely (singly) split epimorphism if and only if 𝑋 is 𝜇-pure in 𝑌 for all matrices 𝜇 of finite columns (single column) over 𝑅 if and only if 𝑋 is finitely (singly) split in 𝑌. Then, in [4], Kaleboğaz called the short exact sequence 𝑋 → 𝑌 → 𝑍 is 𝜇-pure-exact sequence if 𝑢 is 𝜇-pure monomorphism (or 𝑓 is 𝑀-pure epimorphism). 𝜇-pure-exact sequences for all finite matrices 𝜇 over 𝑅 in Mod-𝑅 are coincide with pure-exact sequences.…”

“…𝜇-pure-exact sequences for all finite matrices 𝜇 over 𝑅 in Mod-𝑅 are coincide with pure-exact sequences. In [4], Kaleboğaz also called a short exact sequence finite (single) pureexact sequence if it is a 𝜇-pure exact sequences for all matrices 𝜇 of finite columns (one column) over 𝑅 in Mod-𝑅 and she called the class of all finite (single) pureexact sequences, finite (single) pure-exact structure on Mod-𝑅. and 𝐼 =⊕ 𝑖=1 ∞ 𝐹.…”