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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.04174 |
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| _version_ | 1866913640281538560 |
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| author | Pillay, Anand Rothmaler, Philipp |
| author_facet | Pillay, Anand Rothmaler, Philipp |
| contents | We show that the free module of infinite rank $R^{(κ)}$ purely embeds every $κ$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(κ)}$ whose projectivity is equivalent to left perfectness, which allows to add a `stronger' equivalent condition: $R^{(κ)}$ purely embeds every $κ$-generated flat left $R$-module which is a model of $T$.
We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.
This paper is a condensed version, solely about modules, of our larger work arXiv:2407.15864, with two new results added about cyclically presented modules (Cor.14) and finitely presented cyclic modules (Rem.15). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bass modules and embeddings into free modules Pillay, Anand Rothmaler, Philipp Rings and Algebras Logic We show that the free module of infinite rank $R^{(κ)}$ purely embeds every $κ$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(κ)}$ whose projectivity is equivalent to left perfectness, which allows to add a `stronger' equivalent condition: $R^{(κ)}$ purely embeds every $κ$-generated flat left $R$-module which is a model of $T$. We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules. This paper is a condensed version, solely about modules, of our larger work arXiv:2407.15864, with two new results added about cyclically presented modules (Cor.14) and finitely presented cyclic modules (Rem.15). |
| title | Bass modules and embeddings into free modules |
| topic | Rings and Algebras Logic |
| url | https://arxiv.org/abs/2501.04174 |