First elements for partial-order actions in \(R\)-\(\boldsymbol{Mod}\)

Luis Fernando García-Mora; Hugo Alberto Rincón-Mejía

doi:10.56754/0719-0646.2803.479

Authors

Luis Fernando García-Mora Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico https://orcid.org/0000-0001-6226-0076
Hugo Alberto Rincón-Mejía Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico https://orcid.org/0000-0002-1527-8879

MSC:

16S90, 16S99, 16K99

DOI:

https://doi.org/10.56754/0719-0646.2803.479

Keywords:

R-pr, A-first modules, prime modules, V-rings, left local rings

Abstract

We study notions of primeness in module theory through the lens of order actions on lattices. Given a poset \(\mathcal{P}\) acting on a bounded lattice \(\mathcal{L}\), we introduce \(\mathcal{P}\)-first (and \(\mathcal{P}\)-prime) elements and establish basic permanence properties and examples arising from module-theoretic lattices. Specializing to the natural action of the lattice \(R\text{-}\mathbf{pr}\) of preradicals on the submodule lattice of a left \(R\)-module, we define \(R\text{-}\mathbf{pr}\)-first submodules and \(R\text{-}\mathbf{pr}\)-first modules and prove that, for nonzero modules, they coincide with the BJKN notion of prime module (equivalently, every nonzero submodule cogenerates the module). We then extend the theory to \(\mathcal{A}\)-first and \(\mathcal{A}\)-fully first modules for subclasses \(\mathcal{A} \subseteq R\text{-}\mathbf{pr}\). As applications, we characterize rings for which every nonzero module is \(R\text{-}\mathbf{pr}\)-first, and we describe left semiartinian rings, left local rings, and semisimple homogeneous rings (equivalently, left semiartinian left local \(V\)-rings) in terms of firstness relative to suitable subclasses of preradicals.

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