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On a category of chains of modules whose endomorphism rings have at most $2n$ maximal ideals

Federico Campanini

TL;DR

The paper studies endomorphism rings of objects in ${\mathcal{E}}_n$, the category of right $R$-modules with a fixed chain $0=M^{(0)}\le M^{(1)}\le \cdots \le M^{(n)}=M$, generalizing previous uniserial-module results. It introduces a 2n-class framework built from monogeny/epigeny notions for the factor modules $U^{(i)}=M^{(i)}/M^{(i-1)}$, and shows that for objects with uniserial factors the endomorphism rings are semilocal with at most $2n$ maximal ideals, while the direct-sum structure is governed by these invariants. The main contributions include a precise description of End$_{{\mathcal{E}}_n}(M)$ (Theorem akin to EARY) and a classification of finite direct sums via the 2n invariants (the generalization of the TAMS monogeny/epigeny framework). The methods combine semilocality, type considerations, and factor-category techniques to reduce general cases to the uniserial-factor setting, with a caveat and counterexample when the uniserial assumption is dropped.

Abstract

We describe the endomorphism rings in an additive category whose objects are right $R$-modules $M$ with a fixed chain of submodules $0=M^{(0)}\leq M^{(1)}\leq M^{(2)} \leq \dots \leq M^{(n)}=M$ and the behaviour of these objects as far as their direct sums are concerned.

On a category of chains of modules whose endomorphism rings have at most $2n$ maximal ideals

TL;DR

The paper studies endomorphism rings of objects in , the category of right -modules with a fixed chain , generalizing previous uniserial-module results. It introduces a 2n-class framework built from monogeny/epigeny notions for the factor modules , and shows that for objects with uniserial factors the endomorphism rings are semilocal with at most maximal ideals, while the direct-sum structure is governed by these invariants. The main contributions include a precise description of End (Theorem akin to EARY) and a classification of finite direct sums via the 2n invariants (the generalization of the TAMS monogeny/epigeny framework). The methods combine semilocality, type considerations, and factor-category techniques to reduce general cases to the uniserial-factor setting, with a caveat and counterexample when the uniserial assumption is dropped.

Abstract

We describe the endomorphism rings in an additive category whose objects are right -modules with a fixed chain of submodules and the behaviour of these objects as far as their direct sums are concerned.

Paper Structure

This paper contains 5 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Objects with a semilocal endomorphism ring
  3. i-th monogeny class and i-th epigeny class
  4. Factor categories and finite direct sums
  5. General case

Key Result

Proposition 2.1

Let $M$ be an object of ${\mathcal{E}}_n$, with factor modules $U^{(i)}=M^{(i)}/M^{(i-1)}$, $i=1,\dots,n$. Then: In particular, for every object $M$ of ${\mathcal{U}}_n$, $E_M$ is of finite type.

Theorems & Definitions (32)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Example 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • ...and 22 more