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First elements associated with partial order actions in $R$-Mod

Luis Fernando García-Mora, Hugo Alberto Rincón-Mejía

TL;DR

The paper develops a preradical-based framework to study primeness-inspired module notions in $R$-Mod, introducing $ ext{$ ext{A}$-first}$ and $ ext{$ ext{A}$-fully first}$ modules and linking them to BJKN-prime and to radical/torsion theories. It shows how fully invariant submodules, cogeneration, and socle constructions interact with preradicals through canonical functors like $oldsymbol{eta}$ and $oldsymbol{ ho}$, yielding a unified lattice-theoretic approach. A central result is the equivalence between $R$-pr-first modules and BJKN-prime, with multiple characterizations and structural consequences such as diuniformity of the fully invariant submodule lattice. The paper then extends these ideas to subclasses of preradicals, analyzes the resulting module classes, and derives ring-theoretic consequences, notably characterizations of rings for which every module is $R$-pr-first, tying them to left semiartinian, left local, and $V$-ring properties and illustrating with examples like $oldsymbol{ ext{Z}}_{p^2}$.

Abstract

We explore some concepts of module theory that derive from the notion of primeness, such as first modules, and extend them to more general environments. We also provide descriptions of simple left semiartinian rings, left local rings, semisimple rings, and simple rings in terms of their $\mathscr A$-first modules with respect to a preradical class.

First elements associated with partial order actions in $R$-Mod

TL;DR

The paper develops a preradical-based framework to study primeness-inspired module notions in -Mod, introducing ext{A} and ext{A} modules and linking them to BJKN-prime and to radical/torsion theories. It shows how fully invariant submodules, cogeneration, and socle constructions interact with preradicals through canonical functors like and , yielding a unified lattice-theoretic approach. A central result is the equivalence between -pr-first modules and BJKN-prime, with multiple characterizations and structural consequences such as diuniformity of the fully invariant submodule lattice. The paper then extends these ideas to subclasses of preradicals, analyzes the resulting module classes, and derives ring-theoretic consequences, notably characterizations of rings for which every module is -pr-first, tying them to left semiartinian, left local, and -ring properties and illustrating with examples like .

Abstract

We explore some concepts of module theory that derive from the notion of primeness, such as first modules, and extend them to more general environments. We also provide descriptions of simple left semiartinian rings, left local rings, semisimple rings, and simple rings in terms of their -first modules with respect to a preradical class.
Paper Structure (8 sections, 20 theorems, 1 equation)

This paper contains 8 sections, 20 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preradicals in $R\textit{-Mod}$.
  3. Relationships between preradicals and fully invariant submodules.
  4. $\mathscr{P}$-first elements in a lattice.
  5. $R$-pr-first modules.
  6. First modules relative to a subclass of $\textbf{$R$-pr}$.
  7. The classes of $\mathscr{A}$-first modules and the class of $\mathscr{A}$-fully first modules
  8. Rings for which every module is $\textbf{$R$-pr}$-first.

Key Result

Proposition 1.3

Let $M$ be a nonzero $R$-module. $M$ is $BJKN$-prime if and only if, for all proper submodules $L$ and $N$ of $_R M$, the product $L \bullet_M N$ is a nonzero submodule of $M$.

Theorems & Definitions (70)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3: Bic1 Proposition 2.1
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • proof
  • Definition 2.1
  • Definition 2.2
  • ...and 60 more