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The dual notion of morphic modules over commutative rings

Faranak Farshadifar

TL;DR

This paper defines comorphic $R$-modules as the dual of morphic $R$-modules over a commutative ring and proves a characterisation: $M$ is comorphic iff every submodule $N$ with $M/N$ finitely cogenerated satisfies $N=aM$ and $(N:_RM)=Ra+Ann_R(M)$. It further shows that finite sums of completely irreducible submodules remain completely irreducible, that comorphic modules are co-Hopfian, introduces the co-Bézout notion, proves that under the DAC and $I^M_0(M)\neq 0$ a comorphic module forces $R$ to be morphic and $M$ to be co-Bézout, and develops quasi morphic/comorphic generalizations with implications for cyclic and cocyclic modules. These results extend the dual theory of morphic modules and connect to morphic rings, offering structural insights and potential classifications in module theory over commutative rings.

Abstract

Let R be a commutative ring with identity and M be an R-module. The purpose of this paper is to introduce and investigate the dual notion of morphic modules over a commutative ring.

The dual notion of morphic modules over commutative rings

TL;DR

This paper defines comorphic -modules as the dual of morphic -modules over a commutative ring and proves a characterisation: is comorphic iff every submodule with finitely cogenerated satisfies and . It further shows that finite sums of completely irreducible submodules remain completely irreducible, that comorphic modules are co-Hopfian, introduces the co-Bézout notion, proves that under the DAC and a comorphic module forces to be morphic and to be co-Bézout, and develops quasi morphic/comorphic generalizations with implications for cyclic and cocyclic modules. These results extend the dual theory of morphic modules and connect to morphic rings, offering structural insights and potential classifications in module theory over commutative rings.

Abstract

Let R be a commutative ring with identity and M be an R-module. The purpose of this paper is to introduce and investigate the dual notion of morphic modules over a commutative ring.
Paper Structure (2 sections, 16 theorems, 5 equations)

This paper contains 2 sections, 16 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Proposition 2.4

Let $M$ be a comorphic $R$-module. Then the sum of two completely irreducible submodules of $M$ is a completely irreducible submodule of $M$.

Theorems & Definitions (39)

  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • ...and 29 more