Primitivity Testing in Free Group Algebras via Duality
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915158279847936 |
|---|---|
| author | Seidel, Matan Ernst-West, Danielle Puder, Doron |
| author_facet | Seidel, Matan Ernst-West, Danielle Puder, Doron |
| contents | Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\leq K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$.
As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module.
A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Primitivity Testing in Free Group Algebras via Duality Seidel, Matan Ernst-West, Danielle Puder, Doron Group Theory Rings and Algebras 20E05, 20c07, 13C10 (Primary) 16S34, 20g40 (Secondary) Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\leq K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$. As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module. A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces. |
| title | Primitivity Testing in Free Group Algebras via Duality |
| topic | Group Theory Rings and Algebras 20E05, 20c07, 13C10 (Primary) 16S34, 20g40 (Secondary) |
| url | https://arxiv.org/abs/2502.12885 |