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Gorenstein flat preenvelopes and weakly Ding injective covers

Alina Iacob

Abstract

We consider a (left) coherent ring R. We prove that if the character module of every Ding injective (left) R-module is Gorenstein flat, then the class of Gorenstein flat (right) R-modules, GF, is preenveloping. We show that this is the case when every injective (left) R-module has finite flat dimension. In particular, GF is preenveloping over any Ding-Chen ring.\\ The proofs use the class of weakly Ding injective (left) R-modules, wDI. We show that, when wDI is closed under extensions, the following statements are equivalent:\\ 1. The character module of every Ding injective left R-module is a Gorenstein flat right R-module.\\ 2. The class of weakly Ding injective left R-modules is closed under direct limits.\\ 3. The class of weakly Ding injective modules is covering.\\ The equivalent statements (1)-(3) imply that GF is preenveloping

Gorenstein flat preenvelopes and weakly Ding injective covers

Abstract

We consider a (left) coherent ring R. We prove that if the character module of every Ding injective (left) R-module is Gorenstein flat, then the class of Gorenstein flat (right) R-modules, GF, is preenveloping. We show that this is the case when every injective (left) R-module has finite flat dimension. In particular, GF is preenveloping over any Ding-Chen ring.\\ The proofs use the class of weakly Ding injective (left) R-modules, wDI. We show that, when wDI is closed under extensions, the following statements are equivalent:\\ 1. The character module of every Ding injective left R-module is a Gorenstein flat right R-module.\\ 2. The class of weakly Ding injective left R-modules is closed under direct limits.\\ 3. The class of weakly Ding injective modules is covering.\\ The equivalent statements (1)-(3) imply that GF is preenveloping
Paper Structure (4 sections, 31 theorems, 3 equations)

This paper contains 4 sections, 31 theorems, 3 equations.

Table of Contents

  1. introduction
  2. preliminaries
  3. main results
  4. Gorenstein flat preenvelopes

Key Result

Theorem 1

([22], Theorem 44) The pair$(^\bot \mathcal{DI}, \mathcal{DI})$ is a complete hereditary cotorsion pair over any ring $R$ (in fact, this is a perfect cotorsion pair, i.e. $^\bot{\mathcal{DI}}$ is covering and $\mathcal{DI}$ is enveloping).

Theorems & Definitions (64)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • ...and 54 more