(Strongly) $M-\pazocal{A}-$Injective(Flat) Modules

TL;DR

This paper introduces and characterizes classes of (strongly) $M$-$ ext{A}$-injective and flat modules, exploring their properties, relationships, and special module classes like $ ext{A}$-coherent modules, with a focus on module precovers and preenvelopes.

Contribution

It defines new classes of (strongly) $M$-$ ext{A}$-injective and flat modules, provides their characterizations, and investigates their module-theoretic properties and relationships.

Findings

Characterizations of (strongly) $M$-$ ext{A}$-injective and flat modules.

Relationships between these classes and other module classes.

Results on precovers and preenvelopes of these modules.

Abstract

Let $M$ be a left $R-$module and $\pazocal{A}=\{A\}_{A\in\pazocal{A}}$ be a family of some submodules of $M$. It is introduced the classes of (strongly) $M-\pazocal{A}-\mathrm{injective}$ and (strongly) $M-\pazocal{A}-\mathrm{flat}$ modules which are denoted by $(S) M-\pazocal{A}I$ and $(S) M-\pazocal{A}F$, respectively. It is obtained some characterizations of these classes and the relationships between these classes. Moreover it is investigated $(S) M-\pazocal{A}I$ and $(S) M-\pazocal{A}F$ precovers and preenvelopes of modules. It is also studied $\pazocal{A}$-coherent, $F\pazocal{A}$ and $P\pazocal{A}$ modules. Finally more generally we give the characterization of $S-\pazocal{A}I(F)$ modules where $\pazocal{A}=\{A\}_{A\in\pazocal{A}}$ is a family of some left $R-$modules.