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Second semimodules over commutative semirings

Faranak Farshadifar

TL;DR

The paper extends the notion of second submodules to semimodules over commutative semirings by defining second subsemimodules as those for which, for every $a\in R$, either $aN=S$ or $aN=0$, and exploring their basic properties, including the link that $Ann_R(N)$ is a prime $k$-ideal. It establishes equivalences, stability under $k$-regular maps, and conditions under which secondness follows from or implies prime-ideal structure, particularly in comultiplication/k-comultiplication settings. It further analyzes the structure of second subsemimodules under product semirings, investigates socle constructions, and discusses Noetherian/Artinian implications and the existence of maximal second subsemimodules, aiming to transfer ring-theoretic dual notions to semirings with applications to algebraic structures lacking subtraction.

Abstract

Let R be a semiring. We say that a non-zero subsemimodule S of an R-semimodule M is second if for each a \in R, we have aS = S or aS = 0. The aim of this paper is to study the notion of second subsemimodules of semimodules over commutative semirings.

Second semimodules over commutative semirings

TL;DR

The paper extends the notion of second submodules to semimodules over commutative semirings by defining second subsemimodules as those for which, for every , either or , and exploring their basic properties, including the link that is a prime -ideal. It establishes equivalences, stability under -regular maps, and conditions under which secondness follows from or implies prime-ideal structure, particularly in comultiplication/k-comultiplication settings. It further analyzes the structure of second subsemimodules under product semirings, investigates socle constructions, and discusses Noetherian/Artinian implications and the existence of maximal second subsemimodules, aiming to transfer ring-theoretic dual notions to semirings with applications to algebraic structures lacking subtraction.

Abstract

Let R be a semiring. We say that a non-zero subsemimodule S of an R-semimodule M is second if for each a \in R, we have aS = S or aS = 0. The aim of this paper is to study the notion of second subsemimodules of semimodules over commutative semirings.
Paper Structure (2 sections, 19 theorems, 6 equations)

This paper contains 2 sections, 19 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Second subsemimodules

Key Result

Proposition 2.2

Let $N$ be a subsemimodule of an $R$-semimodule $M$. Then the following are equivalent:

Theorems & Definitions (44)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • Proposition 2.8
  • ...and 34 more