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On momoid graded semihereditary rings

Haneen Falah Ghalib Al-Kharsan, Parviz Sahandi, Nematollah Shirmohammadi

Abstract

In order to study graded left hereditary and left semihereditary rings graded by a cancelation monoid in terms of their modules, we need to revisit graded free, projective, injective, and flat modules and provide graded versions of specific results concerning these modules, like Baer's criterion on injectivity and Lazard's theorem on flatness. Then, among other things, we can give some characterization of graded left hereditary and left semihereditary rings, in particular, of graded-Prüfer and graded-Dedekind domains.

On momoid graded semihereditary rings

Abstract

In order to study graded left hereditary and left semihereditary rings graded by a cancelation monoid in terms of their modules, we need to revisit graded free, projective, injective, and flat modules and provide graded versions of specific results concerning these modules, like Baer's criterion on injectivity and Lazard's theorem on flatness. Then, among other things, we can give some characterization of graded left hereditary and left semihereditary rings, in particular, of graded-Prüfer and graded-Dedekind domains.
Paper Structure (9 sections, 28 theorems, 14 equations)

This paper contains 9 sections, 28 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Graded Rings and modules
  3. Preliminaries
  4. Graded projective, injective, and flat modules
  5. Graded-projective modules
  6. Graded-injective modules
  7. Graded-flat modules
  8. Graded hereditary rings
  9. Graded semihereditary rings

Key Result

Lemma 2.1

Let $I$ be a directed partially ordered set, and $\{M_i,\varphi_{ij}\}$, $\{N_i,\psi_{ij}\}$, and $\{L_i,\theta_{ij}\}$ be direct systems of $\Gamma$-graded left $R$-modules.

Theorems & Definitions (60)

  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • ...and 50 more