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A point model for the free cyclic submodules over ternions

Hans Havlicek, Boris Odehnal, Jaroslaw Kosiorek

Abstract

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T^2, where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.

A point model for the free cyclic submodules over ternions

Abstract

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T^2, where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.

Paper Structure

This paper contains 3 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Final remarks

Key Result

Lemma 1

For any $(u,v)\in\mathbb F^2\setminus\{(0,0)\}$ and $i\in\{1,2\}$ let Then $\gamma(u,v):=\mathbb F q_1(u,v)+\mathbb F q_2(u,v)+\mathbb F r(u,v)$ is a plane. As $(u,v)$ varies in $\mathbb F^2\setminus\{(0,0)\}$ the union of these planes equals $\mathcal{X}\cup\mathop{\mathcal{Y}}$.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof