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Decompositions into a direct sum of projective and stable submodules

Gulizar Gunay, Engin Mermut

Abstract

A module $M$ is {called} stable if it has no nonzero projective direct summand. For a ring $ R $, we study conditions under which $R$-modules from certain classes decompose as a direct sum of a projective submodule and a stable submodule. Over {an arbitrary} ring, modules of finite uniform dimension or finite hollow dimension can be decomposed as a direct sum of a projective submodule and a stable submodule. By using the Auslander-Bridger transpose of finitely presented modules, we prove that every finitely presented right $R$-module over a left semihereditary ring $R$ has such a decomposition. Our main focus in this article is to give examples where such a decomposition fails. We give some ring examples over which there exists an infinitely generated or finitely generated or finitely presented module where such a decomposition fails. Our main example is a cyclically presented module $M$ over a commutative ring such that~$M$ has no such decomposition and $M$ is not projectively equivalent to a stable module.

Decompositions into a direct sum of projective and stable submodules

Abstract

A module is {called} stable if it has no nonzero projective direct summand. For a ring , we study conditions under which -modules from certain classes decompose as a direct sum of a projective submodule and a stable submodule. Over {an arbitrary} ring, modules of finite uniform dimension or finite hollow dimension can be decomposed as a direct sum of a projective submodule and a stable submodule. By using the Auslander-Bridger transpose of finitely presented modules, we prove that every finitely presented right -module over a left semihereditary ring has such a decomposition. Our main focus in this article is to give examples where such a decomposition fails. We give some ring examples over which there exists an infinitely generated or finitely generated or finitely presented module where such a decomposition fails. Our main example is a cyclically presented module over a commutative ring such that~ has no such decomposition and is not projectively equivalent to a stable module.

Paper Structure

This paper contains 4 sections, 17 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Decompositions of Modules with Finite Uniform/Hollow Dimensions
  3. The Auslander-Bridger Transpose and Decompositions of Modules
  4. Examples where the Decomposition Fails

Key Result

Lemma 2.1

If a module $M$ cannot be decomposed as $M=P \oplus N$ where $P$ is a projective submodule and $N$ is a stable submodule, then there exists a sequence $(P_{k})_{k=1}^{\infty}$ of nonzero proper projective submodules of $M$ and a sequence $(N_{k})_{k=1}^{\infty}$ of nonzero proper submodules of $M$ s and so $\operatorname{u.dim}(M)=\infty=\operatorname{h.dim}(M)$ and $M$ contains the infinite direc

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Proposition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 23 more