# Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

## TL;DR

This paper demonstrates a specific associative ring of order 16 where the relations among nonunimodular free cyclic submodules form a geometric structure called a doily, which may have implications for quantum information theory.

## Contribution

It introduces a new connection between algebraic structures of a particular ring and geometric configurations relevant to quantum information.

## Key findings

- Existence of a ring of order 16 with specific submodule relations
- Relations form a structure isomorphic to a generalized quadrangle of order two
- Potential relevance of the geometric structure for quantum information

## Abstract

It is shown that there exists a particular associative ring with unity of order 16 such that the relations between nonunimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.