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Almost Noetherian rings and modules

Xiaolei Zhang

Abstract

In this paper, we investigate the notions of almost Noetherian rings and modules. In details, we give the Cohen type theorem, Eakin-Nagata type theorem, Kaplansky type Theorem and Hilbert basis theorem and some other rings constructions for almost Noetherian rings. In particular, we resolve a question proposed in \cite[9, B. Zavyalov, {\it Almost coherent modules and almost coherent sheaves}, Memoirs of the European Mathematical Society 19. Berlin: European Mathematical Society (EMS), 2025] under a certain condition.

Almost Noetherian rings and modules

Abstract

In this paper, we investigate the notions of almost Noetherian rings and modules. In details, we give the Cohen type theorem, Eakin-Nagata type theorem, Kaplansky type Theorem and Hilbert basis theorem and some other rings constructions for almost Noetherian rings. In particular, we resolve a question proposed in \cite[9, B. Zavyalov, {\it Almost coherent modules and almost coherent sheaves}, Memoirs of the European Mathematical Society 19. Berlin: European Mathematical Society (EMS), 2025] under a certain condition.

Paper Structure

This paper contains 7 sections, 19 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Almost Noetherian modules and their basic properties
  3. Cohen type theorem for almost Noetherian rings and modules
  4. Eakin-Nagata theorem for almost Noetherian rings
  5. Kaplansky type theorem for almost Noetherian rings
  6. Hilbert basis theorem for almost Noetherian rings
  7. Ring constructions for almost Noetherian rings

Key Result

Proposition 2.3

Let $0\rightarrow M\rightarrow N\rightarrow L\rightarrow 0$ be an exact sequence of $R$-modules. Then $N$ is almost Noetherian if and only if $M$ and $L$ are almost Noetherian.

Theorems & Definitions (39)

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