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Formal matrix representations of pseudo-Frobenius and Frobenius rings

Dominik Krasula

TL;DR

The paper develops a formal matrix (Peirce) framework to study pseudo-Frobenius and Frobenius rings, introducing a precise combinatorial criterion to determine when a semiperfect ring carries a given Nakayama permutation $\pi$. It shows how PF/QF blocks appear along cycles of $\pi$, characterises essential socles via pairing bimodules, and describes when corner rings inherit Nakayama structures. A constructive glueing method is then developed to combine indecomposable rings into a larger ring whose Nakayama permutation is the concatenation of the originals, under mild compatibility assumptions on residue fields. As applications, the work yields indecomposable Frobenius rings with non-isomorphic endomorphism rings of simple modules and demonstrates how corner rings along a Nakayama cycle can exhibit Morita dualities and endomorphism-ring isomorphisms, enriching the landscape of PF/QF rings beyond the classical QF/Frobenius case.

Abstract

Rings with Nakayama permutations, pseudo-Frobenius and Frobenius rings in particular, are studied by applying the general theory of formal matrix rings to their Peirce decompositions. A combinatorial criterion is given to decide whether a formal matrix ring with local rings on the diagonal has a prescribed Nakayama permutation. It is shown that a pseudo-Frobenius ring R can be represented as a block matrix ring, where the blocks on the diagonal are pseudo-Frobenius rings corresponding to cycles in the Nakayama permutation of R. All possible supports of such blocks are characterised. In the finite case, their local corner rings are shown to be isomorphic. We characterise local corners of quasi-Frobenius rings as a subclass of rings with a Morita self-duality. The duality contexts between these corners then appear on the shifted diagonal of their formal matrix representations. Using the combinatorial criterion, we give, under mild assumptions, a method of how to glue two rings with a Nakayama permutation. It is then used to construct an indecomposable Frobenius ring with two simple modules whose rings of endomorphisms are not isomorphic.

Formal matrix representations of pseudo-Frobenius and Frobenius rings

TL;DR

The paper develops a formal matrix (Peirce) framework to study pseudo-Frobenius and Frobenius rings, introducing a precise combinatorial criterion to determine when a semiperfect ring carries a given Nakayama permutation . It shows how PF/QF blocks appear along cycles of , characterises essential socles via pairing bimodules, and describes when corner rings inherit Nakayama structures. A constructive glueing method is then developed to combine indecomposable rings into a larger ring whose Nakayama permutation is the concatenation of the originals, under mild compatibility assumptions on residue fields. As applications, the work yields indecomposable Frobenius rings with non-isomorphic endomorphism rings of simple modules and demonstrates how corner rings along a Nakayama cycle can exhibit Morita dualities and endomorphism-ring isomorphisms, enriching the landscape of PF/QF rings beyond the classical QF/Frobenius case.

Abstract

Rings with Nakayama permutations, pseudo-Frobenius and Frobenius rings in particular, are studied by applying the general theory of formal matrix rings to their Peirce decompositions. A combinatorial criterion is given to decide whether a formal matrix ring with local rings on the diagonal has a prescribed Nakayama permutation. It is shown that a pseudo-Frobenius ring R can be represented as a block matrix ring, where the blocks on the diagonal are pseudo-Frobenius rings corresponding to cycles in the Nakayama permutation of R. All possible supports of such blocks are characterised. In the finite case, their local corner rings are shown to be isomorphic. We characterise local corners of quasi-Frobenius rings as a subclass of rings with a Morita self-duality. The duality contexts between these corners then appear on the shifted diagonal of their formal matrix representations. Using the combinatorial criterion, we give, under mild assumptions, a method of how to glue two rings with a Nakayama permutation. It is then used to construct an indecomposable Frobenius ring with two simple modules whose rings of endomorphisms are not isomorphic.

Paper Structure

This paper contains 18 sections, 22 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Formal matrix representations of semiperfect rings
  4. Basic rings
  5. Formal matrix rings
  6. Modules over formal rings
  7. Combinatorial criterion
  8. Socles
  9. Structural properties of formal matrix ring
  10. Main theorem
  11. Essential socle
  12. Combinatorial criterion
  13. Rings whose Nakayama permutation is a cycle
  14. Glueing of rings with respect to their Nakayama permutations
  15. Main Theorem and its aplications
  16. ...and 3 more sections

Key Result

Lemma 5

Let $R$ be a basic formal matrix ring of order $n$ and let $T$ be a simple right $R$-module. Then there exists $i\leq n$ such that $T$ is isomorphic to a module $\epsilon_i \mathfrak{m}_i$ with all proper homomorphisms of module multiplication being zero.

Theorems & Definitions (39)

  • Definition 1
  • Remark 1
  • Example 2
  • Definition 3
  • Remark 2
  • Definition 4
  • Lemma 5
  • Remark 3
  • Lemma 6
  • Lemma 7
  • ...and 29 more