Reduced rank in $σ[M]$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909998173388800 |
|---|---|
| author | Beachy, John A. Medina-Bárcenas, Mauricio |
| author_facet | Beachy, John A. Medina-Bárcenas, Mauricio |
| contents | Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context of $σ[M]$. We study the quotient category of $σ[M]$ modulo the hereditary torsion theory cogenerated by the $M$-injective hull of $M$, when $M$ is a semiprime Goldie module. We prove that this quotient category is spectral. We then consider the hereditary torsion theory in $σ[M]$ cogenerated by the $M$-injective hull of $M/\mathfrak{L}(M)$, where $\mathfrak{L}(M)$ is the prime radical of $M$, and we determine when the module of quotients of $M$, with respect to this torsion theory, has finite length in the quotient category. Finally, we give conditions on a module $M$ with endomorphism ring $S$ under which $S$ is an order in an Artinian ring, extending Small's Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_07196 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reduced rank in $σ[M]$ Beachy, John A. Medina-Bárcenas, Mauricio Rings and Algebras 16D90, 16P50 (Primary), 16P70, 16S50 (Secondary) Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context of $σ[M]$. We study the quotient category of $σ[M]$ modulo the hereditary torsion theory cogenerated by the $M$-injective hull of $M$, when $M$ is a semiprime Goldie module. We prove that this quotient category is spectral. We then consider the hereditary torsion theory in $σ[M]$ cogenerated by the $M$-injective hull of $M/\mathfrak{L}(M)$, where $\mathfrak{L}(M)$ is the prime radical of $M$, and we determine when the module of quotients of $M$, with respect to this torsion theory, has finite length in the quotient category. Finally, we give conditions on a module $M$ with endomorphism ring $S$ under which $S$ is an order in an Artinian ring, extending Small's Theorem. |
| title | Reduced rank in $σ[M]$ |
| topic | Rings and Algebras 16D90, 16P50 (Primary), 16P70, 16S50 (Secondary) |
| url | https://arxiv.org/abs/2201.07196 |