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Relatively big projective modules and their applications to direct sum decompositions

Román Álvarez, Dolors Herbera, Pavel Příhoda

TL;DR

This work extends the theory of countably generated projective modules by developing the framework of relatively big projectives via trace ideals and the monoid $V^*(\Lambda)$. It proves that right noetherian PI-rings satisfy the $(*)$-condition, hence cg projectives are always relatively big, and it extends these results to semiperfect and locally semiperfect algebras over $h$-local domains. The authors introduce rank and genus invariants, $\\Psi$ and $\\Phi$, enabling a detailed classification of projectives and a deeper understanding of when infinite direct sums decompose into finite sums of finitely generated modules. They apply these insights to endomorphism rings of torsion-free modules, providing criteria under which direct sums decompose into finitely generated components, thus linking trace ideals, endomorphism rings, and decomposition theory in broad non-noetherian contexts.

Abstract

Countably generated projective modules that are relatively big with respect to a trace ideal were introduced by P. Příhoda, as an extension of Bass' uniformly big projectives. It has already been proved that there are a number of interesting examples of rings whose countably generated projective modules are always relatively big. In this paper, we increase the list of such examples, showing that it includes all right noetherian rings satisfying a polynomial identity. We also show that countably generated projective modules over locally semiperfect torsion-free algebras over $h$-local domains are always relatively big. This last result applies to endomorphism rings of finitely generated torsion-free modules over $h$-local domains. As a consequence, we can give a complete characterization of those $h$-local domains of Krull dimension $1$ for which every direct summand of a direct sum of copies of a single finitely generated torsion-free module is again a direct sum of finitely generated modules.

Relatively big projective modules and their applications to direct sum decompositions

TL;DR

This work extends the theory of countably generated projective modules by developing the framework of relatively big projectives via trace ideals and the monoid . It proves that right noetherian PI-rings satisfy the -condition, hence cg projectives are always relatively big, and it extends these results to semiperfect and locally semiperfect algebras over -local domains. The authors introduce rank and genus invariants, and , enabling a detailed classification of projectives and a deeper understanding of when infinite direct sums decompose into finite sums of finitely generated modules. They apply these insights to endomorphism rings of torsion-free modules, providing criteria under which direct sums decompose into finitely generated components, thus linking trace ideals, endomorphism rings, and decomposition theory in broad non-noetherian contexts.

Abstract

Countably generated projective modules that are relatively big with respect to a trace ideal were introduced by P. Příhoda, as an extension of Bass' uniformly big projectives. It has already been proved that there are a number of interesting examples of rings whose countably generated projective modules are always relatively big. In this paper, we increase the list of such examples, showing that it includes all right noetherian rings satisfying a polynomial identity. We also show that countably generated projective modules over locally semiperfect torsion-free algebras over -local domains are always relatively big. This last result applies to endomorphism rings of finitely generated torsion-free modules over -local domains. As a consequence, we can give a complete characterization of those -local domains of Krull dimension for which every direct summand of a direct sum of copies of a single finitely generated torsion-free module is again a direct sum of finitely generated modules.

Paper Structure

This paper contains 16 sections, 65 theorems, 58 equations.

Table of Contents

  1. The monoid of infinitely generated projective modules
  2. Relatively big projectives
  3. Finite generation modulo an ideal and the (*)-condition
  4. The monoid of relatively big projective modules
  5. Lifting projectives modules modulo a right T-nilpotent ideal
  6. The case of right noetherian PI-rings
  7. Commutative noetherian rings
  8. One-sided noetherian PI-rings
  9. An application to direct sums of finitely generated modules
  10. Projective versus locally projective modules
  11. Semiperfect rings and locally semiperfect rings
  12. Projective modules over semiperfect rings and their trace ideals
  13. Locally semiperfect algebras over h-local domains
  14. Rank and Genus of countably generated projective modules
  15. Infinite sums of finitely generated projective modules
  16. ...and 1 more sections

Key Result

Lemma 1.5

Let $\Lambda$ be a ring.

Theorems & Definitions (90)

  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Remark 1.7
  • Lemma 1.8
  • Lemma 1.9
  • Lemma 1.10
  • Definition 1.11
  • ...and 80 more