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Valuations on Superrings

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

Abstract

A valuation theory for superrings is developed, extending classical constructions from commutative algebra to the $\mathbb Z_2$-graded and supercommutative setting. We define valuations on superrings, investigate their fundamental properties, and explore the construction of Zariski-Riemann superspaces.

Valuations on Superrings

Abstract

A valuation theory for superrings is developed, extending classical constructions from commutative algebra to the -graded and supercommutative setting. We define valuations on superrings, investigate their fundamental properties, and explore the construction of Zariski-Riemann superspaces.
Paper Structure (23 sections, 31 theorems, 46 equations)

This paper contains 23 sections, 31 theorems, 46 equations.

Table of Contents

  1. Basic definitions
  2. Superrings
  3. Krull's Lemma
  4. Supermodules
  5. Integrally closed superrings
  6. Ordered groups
  7. Valuations and Valuation Pairs
  8. Valuations
  9. Valuation pairs
  10. Maximal pairs
  11. The support of a valuation
  12. $v$-convex superideals
  13. Dominance and Extension of valuations
  14. Dominance
  15. Extensions of valuations
  16. ...and 8 more sections

Key Result

Proposition 1.9

If $R$ is a superring, $\mathfrak{q}$ is a prime ideal of $R$ and $\mathfrak{a}$ is any superideal of $R$ not intersecting $U=R_{\overline{0}}\setminus\mathfrak{q}_{\overline{0}}$, then there exists a prime ideal $\mathfrak{p}$ of $R$ containing $\mathfrak{a}$ and not intersecting $U$ and neither $R

Theorems & Definitions (93)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Proposition 1.9: "Super" Krull's Lemma
  • proof
  • ...and 83 more