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Dedekind Superrings and Related Concepts

Pedro Rizzo, Joel Torres Del Valle, Alexander Torres-Gomez

Abstract

This article investigates the properties of Dedekind superrings, invertible supermodules and projective supermodules within the $\mathbb{Z}_2$-graded framework. Rather than treating these entities as specialized instances of general noncommutative ring theory, we develop them intrinsically within the category of supercommutative superrings. We examine the structural parallels to the classical commutative framework and, more importantly, characterize the fundamental discrepancies that emerge in the $\mathbb{Z}_2$-graded setting. In particular, we show that many hallmark equivalences of classical Dedekind domains-including those involving integral closedness and the coincidence of principal and unique factorization domains-fail to persist in the presence of an odd part.

Dedekind Superrings and Related Concepts

Abstract

This article investigates the properties of Dedekind superrings, invertible supermodules and projective supermodules within the -graded framework. Rather than treating these entities as specialized instances of general noncommutative ring theory, we develop them intrinsically within the category of supercommutative superrings. We examine the structural parallels to the classical commutative framework and, more importantly, characterize the fundamental discrepancies that emerge in the -graded setting. In particular, we show that many hallmark equivalences of classical Dedekind domains-including those involving integral closedness and the coincidence of principal and unique factorization domains-fail to persist in the presence of an odd part.
Paper Structure (17 sections, 36 theorems, 32 equations)

This paper contains 17 sections, 36 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Superrings
  4. Supermodules
  5. Supervector spaces and superalgebras
  6. Associated primes
  7. Noetherian and Finitely Presented supermodules
  8. Krull superdimension and regularity
  9. Invertibility
  10. Invertible supermodules and Fractional superideals
  11. Picard group and Cartier divisors
  12. Projective supermodules
  13. Dedekind superrings
  14. Integral elements over superrings
  15. Regular superrings with even Krull super-dimension one
  16. ...and 2 more sections

Key Result

Proposition 2.8

Let $R$ be a superring and $\mathfrak{p}$ a prime ideal of $R$. Then, $(R_\mathfrak{p}, R_\mathfrak{p}\mathfrak{p})$ is local superring, with $R_\mathfrak{p}\mathfrak{p}=\mathfrak{p}_\mathfrak{p}$. $\square$

Theorems & Definitions (82)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 72 more